The #1 Monster Quest : AI tutor AI Learning Service

Connect With Us on Social Media

Monster Quest | AI tutorMonster Quest | AI tutorMonster Quest | AI tutor

Monster Quest | AI tutor The No.1 Homework Finishing Free App

Geometry and Measurement

Solid Geometry - Properties of Three-Dimensional Shapes (Polyhedra, Cylinders, Cones, Spheres)

Q.01

'Find the locus of the center of the circle C, which is tangent to and circumscribes the circle given by (2) and touches the x-axis.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.02

'Region represented by inequalities'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.03

"Consider a cylinder inscribed in a sphere with a radius of 2, and let its height be 2x. (1) Express the radius a of the cylinder's base in terms of x. (2) Express the volume V of the cylinder in terms of x. (3) Find the maximum value of V. [Hokkaido Institute of Technology]"

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.04

'2019 Shibuya Education Academy Makuhari Middle School 1st Attempt (26) The observer is facing the front of the cliff. The observer faces southeast at Cliff A, southwest at Cliff B, and north at Cliff C.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.05

'Why not measure the elongation of a metal rod with a ruler? Explain briefly in about 20 words.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.06

'According to (1), the layers in the region are dipping from south to north, so in the east-west direction, the layers are almost horizontally stacked. Therefore, on cliff C facing south, each layer appears to be almost horizontal.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.07

'Why are raised floors used to prevent dampness and flooding?'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.08

'Prove that the shape given by the polar equation \ r=\\frac{2}{2+\\cos \\theta} \ is the same as the shape given by the complex number equation \ |z|+\\left|z+\\frac{4}{3}\\right|=\\frac{8}{3} \ and sketch the outline of this shape.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.09

'Equations of 3 Spheres'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.10

'Find the equation of a sphere that satisfies the following conditions: (1) Passes through the points A(1,2,4) and B(-5,8,-2) which are the ends of the diameter. (2) Passes through the point (5,1,4) and touches three coordinate planes.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.11

'There are two spherical surfaces S1:(x1)2+(y2)2+(z+3)2=5S_{1}:(x-1)^{2}+(y-2)^{2}+(z+3)^{2}=5 and S2:(x2)2+y2+(z+1)2=8S_{2}:(x-2)^{2}+y^{2}+(z+1)^{2}=8. Let the intersection of spherical surfaces S1,S2S_{1}, S_{2} be the circle CC. Find:\n(1) The coordinates of the center PP and the radius rr of the circle CC\n(2) The equation of the plane αα containing the circle CC'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.12

'Let the point on the circle C with radius 1 centered at the origin as \\( (\\cos \\theta, \\sin \\theta) \\) be P. Let the circle that is tangent to circle C at point P and also tangent to the y-axis be S, and let the coordinates of its center Q be (u, v). (1) Express u and v in terms of \ \\cos \\theta \ and \ \\sin \\theta \, respectively. (2) Let the area of circle S be denoted by \\( D(\\theta) \\). Determine \\( \\lim_{\\theta \\to \\frac{\\pi}{2}-0} \\frac{D(\\theta)}{(\\frac{\\pi}{2}-\\theta)^{2}} \\).'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.13

'In the coordinate space with point O as the origin, let A(5,4,-2).'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.14

'Consider a cube with vertices O(0,0,0), A(1,0,0), B(0,1,0), C(0,0,1), D(0,1,1), E(1,0,1), F(1,1,0), G(1,1,1) in the coordinate space. Let point P be the trisection point of edge OA in the ratio 3:1, point Q be the division point of edge CE in the ratio 1:2, and point R be the division point of edge BF in the ratio 1:3. The plane passing through points P, Q, R is denoted as α.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.15

'Find the equation of the sphere passing through the points (1,1,1), (-1,1,-1), (-1,-1,0), (2,1,0). Also, determine the coordinates of its center and the radius.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.16

'In tetrahedron ABCD, AB²+CD²=BC²+AD²=AC²+BD², and ∠ADB=90°. Let G be the centroid of triangle ABC.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.17

'Considering a rectangle ABCD in space where the coordinates of point A are (5,0,0) and the coordinates of point D are (-5,0,0), with the length of side AB being 5. Furthermore, both the y-coordinate and z-coordinate of point B are positive, and the length of the perpendicular projection from point B to the xy plane is 3. Please find the coordinates of points B and C.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.18

'Find the area through which the line segment connecting a point on the curve represented by the polar equation r=2(1+costheta)(0leqthetaleqfracpi2) r=2(1+\\cos \\theta)(0 \\leq \\theta \\leq \\frac{\\pi}{2}) and the pole mathrmO \\mathrm{O} passes. Basics 182, Mathematics mathrmC \\mathrm{C} p. 303 Reference'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.19

'Find the equation of a spherical surface passing through the points (0,0,0), (6,0,0), (0,4,0), and (0,0,-8). Also, determine the coordinates of its center and the radius.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.20

'Calculate the number of faces (f), edges (e), and vertices (v) of the polyhedron formed by cutting all the vertices of a regular icosahedron with a plane passing through the midpoints of each edge.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.21

'Chapter 3 Properties of Figures EX ⊕ 91 The figure on the right [1] is a polyhedron obtained by cutting out 8 vertices with a plane passing through the midpoints of each edge of a regular hexahedron. Let this polyhedron be denoted as X. The figure on the right [2] is a polyhedron obtained by cutting out vertices with a plane passing through the midpoints of each edge of polyhedron X. Let this polyhedron be denoted as Y. (1) Determine the number of faces, edges, and vertices of polyhedron X, respectively. (2) Determine the number of faces, edges, and vertices of polyhedron Y, respectively.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.22

'Find the length of the side of a regular tetrahedron inscribed in a sphere of radius 1.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.23

'Find the length of a side of a regular tetrahedron inscribed in a sphere of radius 1.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.24

'Example problem 141 of finding the minimum value of a broken line on a tetrahedron\nGiven a tetrahedron ABCD with AB=BC=CA=8, AD=7. When cos∠CAD=11/14, find the following:\n(1) The length of side CD\n(2) The size of ∠ACD\n(3) For point E on side AC, find the minimum value of BE+ED'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.25

"The figure on the right [1] is a polyhedron obtained by cutting out eight vertices of a regular hexahedron with a plane passing through the midpoints of each edge. Let's call this polyhedron X. The figure on the right [2] depicts the polyhedron obtained by cutting out the polyhedron [1], which we'll call Y."

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.26

'In the cuboid in the diagram to the right, AD=AE=1, EF=√3. (1) Find the edge perpendicular to edge BF.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.27

'Consider a right circular cone with a base radius of 2 and a height of \ \\sqrt{5} \. Let \ \\mathrm{O} \ be the vertex of this cone, and let \ \\mathrm{A}, \\mathrm{B} \ be the two ends of the diameter of the base. Also, let \ \\mathrm{P} \ be the midpoint of the line segment \ \\mathrm{OB} \. Find the shortest distance on the lateral surface of the cone from A to P.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.28

"In the figure on the right, the line AB is tangent to circles O and O' at points A and B, respectively. If the radii of circles O and O' are 5 and 4, and the distance between centers O and O' is 6, find the length of segment AB."

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.29

'Paint each face of a regular tetrahedron and a regular hexahedron with paint. Only one color is painted on each face. Also, rotating it 323 degrees to match the coloring is considered the same. When there are 12 colors, there are 7 ways to paint the faces of the regular tetrahedron such that each face has a different color. When there are 8 colors, there are 億 ways to paint the faces of the regular hexahedron such that each face has a different color.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.30

'The diagram on the right shows a solid figure formed by cutting a rectangular prism on a plane containing the edges DH, BF.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.31

'Each face of a cube must be painted in such a way that adjacent faces have different colors. However, rotations of the cube that result in the same coloring are considered the same. (1) How many ways are there to paint using all 6 different colors? (2) How many ways are there to use all 5 different colors?'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.32

'Please find the number of faces f, edges e, and vertices v of the polyhedron formed by cutting all the vertices of a regular dodecahedron with a plane passing through the midpoints of each edge.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.33

'Use the condition that the sum of opposite angles is 180° to prove that a quadrilateral is inscribed in a circle.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.34

'Find the number of faces (f), edges (e), and vertices (v) of the polyhedron obtained by cutting off all the corners of a dodecahedron through a plane passing through the midpoint of each edge.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.35

"In the figure on the right, the line AB touches the circles O and O' at points A and B respectively. If the radii of circles O and O' are denoted as r and r' (r < r'), and the distance between the centers of the two circles is d, then prove that AB = √(d^2 - (r' - r)^2)."

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.36

'Determine the number of faces f, edges e, and vertices v for the following convex polyhedra:\n(1) A convex polyhedra consisting of 12 regular pentagons and 20 regular hexagons\n(2) A convex polyhedra formed by cutting all the corners with a plane passing through points that trisect each edge of a regular tetrahedron as shown in the figure to the right'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.37

'Light emitted from flashlights and the like spreads in a conical shape, but when illuminated at the right angle, the edge of the illuminated area becomes a parabola. This phenomenon occurs because cutting a cone parallel to its generatrix results in a parabola at the cutting edge.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.38

"Position of Points in Space\nJust like points in a plane are represented by a pair of real numbers, points in space can also be described using coordinates, which consist of a triple of real numbers. Let's consider a point C in space and define three mutually orthogonal number lines at point O as shown in the diagram. These are called the x-axis, y-axis, and z-axis respectively, collectively known as the coordinate axes. Additionally, the point O is called the origin.\nThe plane determined by the x-axis and y-axis is known as the xy-plane, the plane determined by the y-axis and z-axis is the yz-plane, and the plane determined by the z-axis and x-axis is the zx-plane.\nIn a coordinate plane, there are two axes, the x-axis (horizontal) and the y-axis (vertical), but in coordinate space, the z-axis (height) is added. These three axes together are referred to as the coordinate plane."

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.39

'Please answer (A to C) that fits in the following table.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.40

'By passing a plane through the midpoints of each edge of the regular dodecahedron and cutting all the vertices, determine the number of faces f, edges e, and vertices v of the resulting polyhedron with 21 faces.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.41

"For a convex polyhedron with 8 vertices and 6 faces, let's find the number of edges."

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.42

'Calculate the number of faces f, edges e, and vertices v of the convex polyhedron formed by cutting all the vertices of a regular octahedron with a plane passing through the points that trisect each edge.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.43

'Choose the term from the following mathematical terms that corresponds to regular polyhedron.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.44

'Prove that in a right square pyramid A-BCDE where all edges have equal length, when the midpoint of edge AD is denoted as M, edge AD is perpendicular to the plane MEC.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.45

'In the pentagonal prism ABCDE-FGHIJ shown in the diagram, answer the following questions.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.46

"In problems related to polyhedra, Euler's polyhedron theorem is used to clarify the relationship between vertices, edges, and faces."

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.47

'The Sine Rule is a theorem that expresses the relationship between the sine of the three internal angles of a triangle and the lengths of its three sides. To prove this theorem, the theorem of the inscribed angles learned in junior high school is used.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.48

'For a right circular cone inscribed in a sphere of radius 1, find the height, base radius, and lateral area to maximize.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.49

'Imagine cutting a rolled paper tube diagonally. How do you think the edge will look when the paper is unfolded? Here, assume the radius of the base is 1, and the angle between the cut edge and the base is π/4 (=45 degrees).'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.50

'In the right pyramid O-ABCD, with the length of one side of the base being 2a and the height being a. Find the following:\n(1) The length of the perpendicular line AE drawn from vertex A to edge OB\n(2) For point E in (1), find the measure of angle AEC and the area of triangle AEC'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.51

'To facilitate the calculation of the area by dividing'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.52

'When all vertices of a regular icosahedron W with side length 1 are on the surface of a sphere S, answer the following questions. A regular icosahedron has all faces congruent equilateral triangles, with each vertex shared by 5 triangles.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.53

'Mathematics I'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.54

'(1) By the cosine rule'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.55

'Consider a right triangular prism T with base triangle ABC where AB=2, AC=3, and BC=t(1<t<5). A right triangular prism is defined as a prism where all edges are perpendicular to the base. Furthermore, suppose a sphere S is contained inside T and is tangent to all faces of T.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.56

'Problem about regular tetrahedron'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.57

'Given that the height of the right triangular prism is 4, let the radius of the sphere be r, then 0 < r ≤ 2'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.58

'Tetrahedron and Sphere'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.59

'As shown in the figure, three points D, E, and F are taken outside △ABC in such a way that △ABD, △BCE, and △CAF form equilateral triangles. Let S be the area of △ABC, and the lengths of the three sides be BC=a, CA=b, AB=c. Answer the following questions: (1) Let ∠BAC=θ, express sinθ in terms of b, c, and S, and express cosθ in terms of a, b, and c. (2) Express DC² in terms of a, b, c, and S. It is permissible to use the general identity cos(60°+θ)=\\frac{cosθ-√3 sinθ}{2}. (3) Let the average area of the three equilateral triangles be T, express DC² in terms of S and T.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.60

'Find the length of one side of a regular tetrahedron inscribed in a sphere of radius 1.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.61

'There is a right circular cone with a base radius of 2 and a height of √5. Let O be the vertex of this cone, A and B be the endpoints of the base diameter. If the midpoint of segment OB is P, what is the shortest distance from A to P on the lateral surface?'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.62

'Finding the minimum value of a broken line on tetrahedron Example 141\nThere is a tetrahedron ABCD with AB=BC=CA=8, AD=7. When cos∠CAD=11/14, find the following:\n(1) The length of edge CD\n(2) The size of ∠ACD\n(3) For a point E on edge AC, find the minimum value of BE+ED\nBasics 121,137'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.63

'Example 127 Measurement Problem (Space)\nAs shown in the diagram on the right, a utility pole stands perpendicular to a plane containing points A, B, and C. When viewed from points A and B, the top of the pole D has angles of elevation of 60° and 45° respectively. Given that the distance between A and B is 6m, and ∠ACB = 30°, find the height of the pole CD. Assume no considerations for eye height.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.64

"Tetrahedron and Sphere\nLet's consider a tetrahedron ABCD with edge length a.\n(1) Express the radius R of the sphere circumscribed about the tetrahedron in terms of a.\n(2) Express the radius r of the sphere inscribed in the tetrahedron in terms of a."

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.65

'Find the length of the edge of a regular tetrahedron inscribed in a sphere of radius 1.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.66

'There is a right circular cone with a base radius of 2 and a height of √5. Let O be the vertex of this cone, A and B be the two ends of the diameter of the base, and P be the midpoint of segment OB. Find the shortest distance from A to P on the lateral surface.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.67

'In space, a point is typically defined by its coordinates and its distance to the origin O.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.68

'Find the equation of a spherical surface with center at point (a, b, c) and radius r.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.69

'Find the equations of the following spheres.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.70

'Translate the coordinates of the center and the radius in order'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.71

'Find the real number a, and when the point P moves over the entire sphere S,'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.72

'As shown in the diagram, let S, T, U be the points of intersection of the plane z=t (0<t<2/3) with the circumference of disk D and the line segment CQ.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.73

'Equation of a sphere'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.74

'Practice: Given the plane alpha:x2y+2z+3=0 \\alpha: x-2 y+2 z+3=0 and two spheres S1:(x1)2+(y2)2+(z+3)2=5 S_{1}:(x-1)^{2}+(y-2)^{2}+(z+3)^{2}=5 , S2:(x2)2+y2+(z+1)2=8 S_{2}:(x-2)^{2}+y^{2}+(z+1)^{2}=8 . Find the following. (1) The equation of a sphere passing through the origin that includes the intersection of the plane alpha \\alpha and the sphere S1 S_{1} (2) The equation of a plane that contains the circle C C of intersection between the spheres S1 S_{1} and S2 S_{2} , along with the coordinates of the center P P of circle C C and the radius r r '

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.75

'What is the shape of the triangle ABC formed by the three points A(4,7,2), B(2,3,-2), C(6,5,-6)?'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.76

'The perpendicular dropped from the center C(a, b, c) of the sphere to the xy-plane passes through the center (5/6, 5/6, 0) of the circle. Therefore, the x and y coordinates of points C and the center of the circle are both equal to a=5/6, b=5/6. Additionally, the radius of the sphere S is OC = √(〖(5/6)〗^2+〖(5/6)〗^2+c^2) = √(c^2 + 25/18). Thus, the equation of sphere S is (x-5/6)^2 + (y-5/6)^2 + (z-c)^2 = c^2 + 25/18. When the point (t+2, t+2, t) lies on sphere S, (t+2-5/6)^2 + (t+2-5/6)^2 + (t-c)^2 = c^2 + 25/18, which simplifies to 9t^2 - 2(3c-7)t + 4 = 0. The necessary and sufficient condition for the line l to have points in common with sphere S is that the quadratic equation for t (1) has real solutions. Therefore, let D be the discriminant, and solving for D≥0 leads to (3c-1)(3c-13)≥0. From this, it follows that c≤1/3 or c≥13/3. Hence, the conditions that a, b, c must satisfy are a=b=5/6 and (c≤1/3 or c≥13/3).'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.77

'Assuming a>0. Find the following for the sphere passing through points O(0,0,0), A(0, a, a), B(a, 0, a), C(a, a, 0) with equation 54.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.78

'Here we have obtained another solution.\n\n\\overrightarrow{\\mathrm{AP}}=(x-1, y, z)\n\n\\overrightarrow{\\mathrm{BP}}+2\\overrightarrow{\\mathrm{CP}}=(x, y-2, z)+2(x, y, z-3)=(3x, 3y-2,3z-6)\n\nTherefore, \\overrightarrow{\\mathrm{AP}} \\cdot(\\overrightarrow{\\mathrm{BP}}+2 \\overrightarrow{\\mathrm{\\mathrm{CP}}})=0 leads to (x-1) \\times 3x + y \\times (3y-2) + z \\times (3z-6)=0\n\nHence \\quad x^{2}-x+y^{2}-\\frac{2}{3} y+z^{2}-2z=0 Therefore \\left(x-\\frac{1}{2}\\right)^{2}+\\left(y-\\frac{1}{3}\\right)^{2}+(z-1)^{2}=\\frac{1}{4}+\\frac{1}{9}+1 that is \\quad\\left(x-\\frac{1}{2}\\right)^{2}+\\left(y-\\frac{1}{3}\\right)^{2}+(z-1)^{2}=\\frac{49}{36} \n\nThus, the set of points P is a spherical surface with center at \\left(\\frac{1}{2}, \\frac{1}{3}, 1\\right) and radius \\frac{7}{6} .'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.79

'Explain the properties of a parabola and deduce the properties of a point P on the parabola.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.80

'(1) Let the center of sphere S1 S_{1} be mathrmO1(1,2,3) \\mathrm{O}_{1}(1,2,-3) with radius r1=sqrt5 r_{1}=\\sqrt{5} . The distance between plane alpha \\alpha and the center mathrmO1 \\mathrm{O}_{1} of sphere S1 S_{1} is frac12cdot2+2cdot(3)+3sqrt12+(2)2+22=2timessqrt5>2 \\frac{|1-2 \\cdot 2+2 \\cdot(-3)+3|}{\\sqrt{1^{2}+(-2)^{2}+2^{2}}}=2 \\times \\sqrt{5}>2 , thus the plane alpha \\alpha intersects with the sphere S1 S_{1} . Therefore, the common points between plane alpha \\alpha and sphere S1 S_{1} satisfy the following equation:\n\\[4pt]\n\\(k(x2y+2z+3)+(x1)2+(y2)2+(z+3)25=0k(x-2 y+2 z+3)+(x-1)^{2}+(y-2)^{2}+(z+3)^{2}-5=0\\)\nEquation (1) represents the sphere. Since it passes through the origin, substituting x=y=z=0 x=y=z=0 , we get\n\\[4pt]\n\$3 k+1+4+9-5=0$\\nTherefore, k=3 k=-3 , substituting k=3 k=-3 into (1), we get\n\\[4pt]\n\\(3(x2y+2z+3)+(x1)2+(y2)2+(z+3)25=0-3(x-2 y+2 z+3)+(x-1)^{2}+(y-2)^{2}+(z+3)^{2}-5=0\\)\nSimplifying, the resulting equation is x2+y2+z25x+2y=0 x^{2}+y^{2}+z^{2}-5 x+2 y=0 '

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.81

'A regular hexagon ABCDEF with side length 1 is given. When point P moves on edge AB and point Q moves on edge CD independently, find the area through which point R, dividing the segment PQ in the ratio 2:1, can pass.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.82

'(2) Since the sphere is tangent to each coordinate plane and passes through the point (5, -1, 4), the radius is denoted as r, and the coordinates of the center are (r, -r, r). Therefore, the equation of the sphere is'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.83

'Find the equation of the plane passing through the point (-1,2,3) and perpendicular to the line given by (4)(x-2)=(y+1)(-3)=z-3'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.84

'Prove that the points where line segments AG_A, BG_B, CG_C, and DG_D are internally divided in the ratios 3:1 coincide for the tetrahedron ABCD with centroids G_A, G_B, G_C, and G_D for the triangles BCD, ACD, ABD, and ABC respectively.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.85

'Let the equation of the sphere be x2+y2+z2+2x4y+4z=16x^{2}+y^{2}+z^{2}+2x-4y+4z=16, and the plane alpha:6x2y+3z=5\\alpha: 6x-2y+3z=5 intersecting it forms a circle CC. Find the coordinates of the center and the radius of this circle.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.86

'Let the surface obtained by rotating the line segment AB connecting the points A(0,0,2) and B(1,0,1) around the z-axis be denoted by S. As the points P on S and Q on the xy-plane move such that PQ = 2, let K be the range that the mid-point M of the line segment PQ can pass through. Find the volume of K.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.87

'Equation of a sphere'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.88

'Find the coordinates of the vertices of the rectangular prism OABC-DEFG on the right, except for point O.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.89

'Consider a circle with a radius of 1 centered at the origin on the xy-plane in coordinate space. Let S be the cone (including its interior) with this circle as its base and the point (0,0,2) as its vertex. Also, consider the point A(1,0,2).'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.90

'(2) Let point B pass through and let α be the plane perpendicular to the x-axis. On the plane α, with point C(1,0,0) as the center and radius CB = √(3²+4²) = 5 being on the circle, let R be a moving point on the circle. Then, CB = CR, QB = √(QC² + CB²), QR = √(QC² + CR²). Therefore, QB = QR, so D(1,0,-5). Then, AQ + QB = AQ + QD ≥ AD. Since points A, Q, and D are on the zx plane, the minimum value of AQ + QD occurs when Q is on the line AD. Therefore, the minimum value of AQ + QB is AD = √((1-2)² + (0-0)² + (-5-3)²) = √65.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.91

'In the cuboid ABCD-EFGH, prove that the midpoints of edges FB, BC, CD, DH, HE, and EF are all in the same plane.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.92

'Consider a triangle ABC with three vertices A, B, C on curve K: y=1/x. Prove that the orthocenter H of triangle ABC lies on curve K.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.93

'Find the volume of the part in coordinate space where the distance to the x-axis, y-axis, and z-axis are all less than or equal to 1.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.94

'There is a circle C with radius 5 centered at the origin O on the coordinate plane. Let n=2 or n=3, and consider a circle C_{n} with radius n that is tangent to circle C and rotates without sliding. There is a point P_{n} on circle C_{n}. Initially, the center O_{n} of circle C_{n} is at (5-n, 0), and point P_{n} is at (5,0), with the center of circle C_{n} rotating counterclockwise n times inside circle C to return to its original position. Let the point of tangency between circles C and C_{n} be S_{n}, and let the angle formed by segment OS_{n} with the positive direction of the x-axis be t.\n(1) Express the coordinates of point P_{n} in terms of t and n.\n(2) Show that the curves described by point P_{2} and point P_{3} are the same.\n[Osaka University]'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.95

'In coordinate space, find the volume of the part where the distance to the x-axis, y-axis, and z-axis are all less than or equal to 1.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.96

'In coordinate space, find the volume of the region where the distance to both the x-axis and y-axis is less than or equal to 1.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.97

'Find the area of the region on the plane z = t (-1 ≤ t ≤ 1) where the distances to the x-axis and y-axis are both less than or equal to 1.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.98

'Let the polar coordinates of point A be (10,0), and let Q be any point on the circle C with diameter formed by the line segment connecting pole O and point A. Draw a perpendicular from pole O to the tangent of circle C at point Q, let the polar coordinates of point P be (r,θ), find the polar equation of its trajectory. Here, 0 ≤ θ < π.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.99

'Example 54: Equation of a Sphere (2)'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.00

'Exercise 1: Prove that the sum 1/OP^2 + 1/OQ^2 is constant when drawing two perpendicular half-lines from the center O of the ellipse to the intersection points P and Q.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.01

"Points O, A', B' are on the xy-plane, so the figure formed by the intersection of the spherical surface S and the xy-plane is a circle passing through O, A', B'."

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.02

'Equation of a sphere - a sphere of radius r centered at point (a, b, c) (x-a)^{2}+(y-b)^{2}+(z-c)^{2}=r^{2}'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.03

'Let \\( \\mathrm{A}(0,2,0) \\) be a point and \\( \\vec{d}=(1,1,-2) \\) be parallel to the line \ \\ell \.\n(1) Find the coordinates of the intersection point of the line \ \\ell \ and the plane \ 2x-3y+z=0 \.\n(2) Find the length of the segment cut by the line \ \\ell \ on the sphere \\( (x-4)^{2}+(y-2)^{2}+(z+4)^{2}=14 \\).'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.04

"(2) When t ranges over all real numbers, let the line determined by the point (t+2, t+2, t) in the xyz space be denoted by l. Given that the sphere S with center at C(a, b, c) passes through the points O(0,0,0), A'(2,1,0), B'(1,2,0) and shares a point with the line l, find the conditions for a, b, c. [Hokkaido University]"

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.05

'In space, the set of points that are a constant distance r from a fixed point C is called a sphere with center C and radius r.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.06

'Explain how the intersection of a cone cut by a plane that does not pass through its vertex forms a second-order curve, and describe the cases for an ellipse, hyperbola, and parabola.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.07

'In tetrahedron ABCDABCD, let GAG_A, GBG_B, GCG_C, GDG_D be the centroids of triangles BCDBCD, ACDACD, ABDABD, ABCABC respectively. Prove that the points where the segments AGAAG_A, BGBBG_B, CGCCG_C, DGDDG_D divide in the ratio 3:13:1 coincide.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.08

'There are four points A(4,0,0), B(0,8,0), C(0,0,4), D(0,0,2).'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.09

'Find the coordinates of the intersection points of the line passing through points A(3,1,-1) and B(-2,-3,2) with the xy-plane, yz-plane, and zx-plane.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.10

'(2) The plane ax + (9-a)y - 18z + 45 = 0 touches a spherical surface centered at (3, 2, 1) with radius √5. Find the value of the constant a.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.11

'Given that the center is (1, -3, 2) and the sphere passing through the origin intersects with the plane z=k to form a circle with a radius of √5. Find the value of k.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.12

'Learned about vectors in the plane, now learning the basics of vectors in space, and understanding the equations of shapes (lines, spheres, etc.) in space coordinates.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.13

'Find the center coordinates and radius of the circle formed by the intersection of a spherical surface with center (-1,5,3) and radius 4 and the plane x=1.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.14

'Find the equation of the following sphere.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.15

'Let \\( \\mathrm{A}(0,3,0), \\mathrm{B}(0,-3,0) \\) be the endpoints of a diameter of a spherical surface \ S \ in the coordinate space. When the point \\( \\mathrm{P}(x, y, z) \\) moves on the surface \ S \, find the maximum value of \ 3x+4y+5z \. Also, determine the coordinates of P at that point.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.16

'For the points A(2,-1,3), B(5,2,3), C(2,2,0), prove that: (1) The triangle with vertices A, B, C is an equilateral triangle. (2) If the three vertices of a regular tetrahedron are A, B, C, find the coordinates of the fourth vertex D.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.17

'Find the coordinates of the intersection points between the line passing through points A(2,4,0) and B(0,-5,6) and the sphere with center at (0,2,0) and radius 2.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.18

'Given that the center of the sphere is (1, -2, 3a) and the radius is √13, when this sphere intersects the xy-plane, it forms a circle with a radius of 2. Find the value of a. Also, determine the coordinates of the center of this circle.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.19

'(1) Find the equations of the shapes formed by the intersection of a sphere with center (-1,3,2) and radius 5 with the xy-plane, yz-plane, and zx-plane.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.20

'In the regular pentagon ABCDE with side length of 1, let AB be vector b and AE be vector e.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.21

'In the complex plane, let points A and B represent -1+2i and 3+i respectively. If AB is one side of a square, find the complex numbers representing the vertices C and D of the square ABCD.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.22

'Consider the 3 points O(0,0,0), A(2,0,1), B(0,1,2). Let point P(x,y,z) move in such a way that |PO|=|PA|=|PB|.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.23

'(2) The sphere with center (1,-2,3a) and radius sqrt(13) intersects the xy-plane to form a circle with a radius of 2. Find the value of a. Also, find the coordinates of the center of this circle.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.24

'Find the equations of the shapes formed by the sphere with center (-1, 3, 2) and radius 5 intersecting the xy-plane, yz-plane, and zx-plane.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.25

'For points A(2,-1,3), B(5,2,3), C(2,2,0): (1) Prove that triangle ABC is an equilateral triangle. (2) If A, B, and C are the three vertices of a regular tetrahedron, find the coordinates of the fourth vertex D.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.26

'Let a>0. Find the following for the sphere passing through points O(0,0,0), A(0,a,a), B(a,0,a), C(a,a,0): (1) Coordinates of the center and radius (2) Equation of the intersection with the zx plane'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.27

'Given that a sphere with center (2, -3, 4) and radius r intersects the xy-plane to form a circle with a radius of 3. Determine the value of r.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.28

'Next, consider a regular octahedron and a sphere tangent to all of its faces, and imagine a cross-section cut through a plane containing the points of contact, as shown in the diagram on the right [2]. If the radius of the sphere is r, then the area of the right-angled triangle in the mesh section is'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.29

'Explain how to find the height of a tetrahedron.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.30

'Assuming the existence of a cube as shown in the right figure, the path from A to D is a permutation of 3 to the right, 2 up, and 1 up,'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.31

'Prove the following for the tetrahedron ABCD:\n1. Let M be the midpoint of edge AB.\n(A) Edge AB is perpendicular to plane CDM.\n(T) Edge AB is perpendicular to edge CD.\n2. Let the midpoints of edges BC, AC, AD, and BD be P, Q, R, S respectively, then the quadrilateral PQRS is a square.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.32

'Determine the number of faces f, edges e, and vertices v of the polyhedron formed by cutting all the vertices of a regular icosahedron with a plane passing through the midpoints of each edge.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.33

'Calculate the minimum number of colors required for a regular hexahedron.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.34

'Please calculate the shortest distance in the unfolded diagram.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.35

'Prove: A is the Foot of Perpendicular of Tetrahedron.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.36

'Find the number of faces f, edges e, and vertices v of the polyhedron formed by cutting all corners through a plane passing through the midpoints of each edge of a regular dodecahedron.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.37

'By the converse of the secant theorem, DA is a tangent to the circle passing through points A, E, and F.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.38

'Consider the tetrahedron ABCD in space. Prove that there exists a spherical surface that passes through all 4 vertices A, B, C, D.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.39

'One wants to color each face of a cube such that neighboring faces have different colors. However, rotations of the cube that result in the same coloring are considered the same.'

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.40

Find the equation of the sphere with its center at the origin and radius r.

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.41

Let a be a real number. In the xyz space, consider the four points A(0, a, 4), B(-2, 0, 3), C(1, 0, 2), and D(0, 2, 3), and place a light source at point P(1, 0, 6). (1) The coordinates of the shadow of point A on the xy plane created by the light source are (アイ, ウ a, 0). (2) The shadow of the triangle BCD on the xy plane created by the light source is also a triangle. The coordinates of the vertices of this triangle are 力 > ク.

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.42

15 (2) \( (x-1)^{2}+(y-1)^{2}+(z-1)^{2}=1 \), \( (x-3)^{2}+(y-3)^{2}+(z-3)^{2}=9 \)

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.43

Intersection of a sphere and planes The intersection of the sphere \( (x+1)^{2}+(y-4)^{2}+(z-2)^{2}=3^{2} \) with the following planes is a circle. Find its center coordinates and radius. (1) xy x y plane (2) yz y z plane (3) plane y=4 y=4

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.44

A sphere centered at the origin with radius r x2+y2+z2=r2 x^{2}+y^{2}+z^{2}=r^{2}

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.45

(1) Find the coordinates of the center and the radius of the sphere x2+y2+z24x4y2z+5=0 x^{2}+y^{2}+z^{2}-4 x-4 y-2 z+5=0 . (2) Find the equation of the sphere passing through the points \( (2,0,0),(0,2,0),(0,0,2),(2,2,2) \).

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.46

Find the equation of a sphere centered at the point (a, b, c) with radius r \(\ (x-a)^{2}+(y-b)^{2}+(z-c)^{2}=r^{2}\)

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.47

Equation of the sphere Find the equations of the following spheres. (1) A sphere centered at the point \( (3,-2,1) \) with radius 2 (2) A sphere centered at the origin that passes through the point \( (2,1,-3) \) (3) A sphere with endpoints of the diameter at points \( \mathrm{A}(5,3,-2) \) and \( \mathrm{B}(-1,3,2) \)

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.48

The intersection of the sphere \( (x-2)^{2}+(y+3)^{2}+(z-5)^{2}=10 \) with the following planes is a circle. Find the coordinates of the center and the radius of the circle. (1) yz-plane (2) zx-plane (3) plane z=3 z=3

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.49

Equation of a sphere (general form)

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.50

Find the equations of the spheres as follows. (1) A sphere centered at the origin with radius 222 \sqrt{2} (2) A sphere centered at point A(6,5,-3) passing through point B(2,4,-3) (3) A sphere with endpoints A(-1,4,9) and B(7,0,1) of its diameter

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.51

Equation of a sphere in coordinate space

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.52

Find the equation of the sphere as follows.

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.53

Shapes in coordinate spaces: Intersection of a sphere and a plane

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord

Q.54

Drop a perpendicular OH \mathrm{OH} from the origin O \mathrm{O} to the plane ABC \mathrm{ABC} determined by the three points \( \mathrm{A}(2,0,0), \mathrm{B}(0,1,0), \mathrm{C}(0,0,-2) \). Find the coordinates of point H \mathrm{H} and the length of segment OH \mathrm{OH} .

A. ...

Ask Monster Quest : AI tutor for answer!
Join our Discord