60. Show that the distance between two points P1(x1, y1) and P2(x2, y2) is invariant under rotation. That is, show that the distance between P1(x1, y1) and P2(x2, y2) equals the distance between P1(xí, yi) and P2(x2, y2).

Q: 4. Trapezoid JKLM with vertices J(2, 1) , K(5, 1), L ^ n * (8, - 4) and M(1,-4);90 ° clockwise…

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Q: Show that the points A = (1, 2), B = (1,0), and C = collinear and express P = (-7,0) in barycentric…

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Q: Find a parametrization for the line segment beginning at P,(4, - 7,- 7) and ending at P2(0, - 7,2).

A: The line segment beginning at P1<4, -7, -7> and ending at P2<0, -7, 2>

Q: Draw a figure that shows the image of the triangle with vertices (0, 0), (1, 0), and (1/2, 1) under…

A: It is given that the vertices of the triangle are 0, 0, 1, 0 and 12, 1. We have to sketch the graph…

Q: 3. Sketch the points determined by the following. (a) Iz+1-i = 2 %3D (b) Im(2z +i) +1=0

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Q: Find a parametrization of the line through P = (4, 9, 8) perpendicular to the yz-plane.

A: Equation of Linethe Line through P =(x0,y0,z0) in the diresction of vector v→=(a,b,c)is defined…

Q: d) If S is projected onto the x - y plane, what is the shape of the resulting projection?

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Q: Find a parametric representation of the line through the points P(4,1,2) and Q(3,5,7)

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Q: express in polar coords with: Cりり me point whose Ty-Coords are a.r>o and OsoL2

A: Introduction: The polar coordinates r,θ of rectangular coordinates (x,y) in two-dimensional…

Q: 8. Find a parametric representation of the line through the point P(1,0,4) and perpendicular to the…

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Q: Find the image of the point A(-2, 4, 5) under a rotation of 60 degrees In the plane x = 2 about the…

A: We need to find image of given point.

Q: 6. Given a rule x2 + y2 = 9, its image rule x2 + y2 – 2x + 6y + 1 = 0 under a translation, state a…

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Q: Consider AXYZ with vertices X(4, 7), Y(8, 5). and Z(6, 3). Follow the steps to the left to find the…

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Q: Graph the image of AUVW after the following glide reflection: Translation (x, y) ► (x - 15, y)…

A: We first find the coordinates of U,V and W U=(3,6), V=(6,3), W=(12,6) Then we apply (x,y)→(x-15,y)…

Q: Sketch the following planes in the window [0, 5] × [0, 5] × [0, 5]. The plane parallel to the…

A: First obtain the plane that parallel to the xz-plane containing the point (1, 2, 3).…

Q: Find parametric equation of the line joining points P{(2,4, –1) and P2(5,0, 7). Where does it…

A: To do this we need the vector V→ that will be parallel to the line. This can be any vector as long…

Q: Suppose that C = {z:|2| = 1} is the unit circle, followed in the positive (anticlockwise) direction.…

A: Given: C=z:z=1is the unit circle  f(z)=∫cdz4+3z

Q: Find the coordinates of the reflection without using the coordinate plane. Q(-3,5) over the y-axis.

A: Since you have asked multiple questions in a single request, we will be answering only the 1st…

Q: Parametrize the line segment joining the points P(1, 2, 0) and Q(1, 3, -1).

A: We know that equation of line passing through the points (x1,y1,z1) and (x2,y2,z2) is

Q: If 1x, y2 are the coordinates of a point P in the xy-plane, then x is called the _________  of P and…

A: If (x,y) are the coordinates of a point P in the xy-plane, then x is called the _________  of P and…

Q: Formulate an verify Green theorem when [P, Q] = [2yx, y² + 2x] and the curve y is the positively…

A: According to the given information it is required to formulate Green theorem. When

Q: Prove that the circles x² + y2 + z2 - 2x + 3y + 4z - 5 = 0, 5y + 6z +1 = 0 x² + y2 + z2 - 3x - 4y +…

A: We will use equation of sphere passing through sphere S  and plane P  S+λP =0

Q: 7. If 1x, y2 are the coordinates of a point P in the xy-plane,then x is called the__________________…

A: Given that, x,y are the coordinates of a point P in the xy-plane. The objective is to fill in the…

Q: Find a parametric representation of the line through the point P(3,1,-4) and perpendicular to the…

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Q: Compute the angle between the two planes 2x + 3y + 7z = 2 and 4x - 2y + 2z = 4.

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Q: with the guldin's proposition 1 get the contents of the rotating object that occurs if the land…

A: As per the question we are given a 2d region bounded by the curve : y = 6 - 3x - x2 and the straight…

Q: 9. Find the angle between the two planes 2x + 3y + z = 3 and x + 3y + 2z = -3

A: Given,           2x+3y+z=3    and   x+3y+2z=-3We know that,           Suppose the two equation be…

Q: Graph the image of AUVW after the following glide reflection: Translation (x, y) (x- 15, y)…

A: Given: The △UVW has done two transformations 1) x,y→x-15,y 2) Reflection across x-axis

Q: 2. Use analytic geometry to verify that quadrilateral JKLM is a parallelogran 1C). YA K(-1,1) 2 -2 0…

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Q: What is the image of the point (4, 0) after a rotation of 90 counterclockwise about the origin

A: Under rotation 90° counter clockwise about the origina point (x,y)→(-y,x)Hence (4,0)→(0,4)

Q: Region A is formed by bounding the lines y= T, y = 6, and the y-axis. Determine which pair %3D of…

A: Region A is formed by bounding the lines y=x, y=6 and y-axis. We have to determine which pair of…

Q: 5. Calculate the distance between the point (3, 1, -2) and the plane 2x + y-2+1. O 10 o -10 V14 O 10…

A: The equation of the plane is  2x+y-z+1 = 0 The point is x0,y0,z0 =3,1,-2 To evaluate : The distance…

Q: Suppose we have: u = and v = . Based on their relative positions in the xy-plane, in which…

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Q: find the pole and resuidue of 3z + 1 (z + 1)(2z – 1)

A: Poles and residue of the given function are given below ;

Q: 5. Which of the Two-Reflection Theorems for Rotations and Translations could apply when the…

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Q: Find the coordinates of the reflection without using the coordinate plane. P (4,6) over the x-axis.

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Q: Consider the projection from the origin of the straight line y + x = 2 on a straight line y + 2x =…

A: Graph formed by

Q: Which of the following points is the central projection of the point (4, -8, 14) onto the plane x=2?…

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Q: 1) Find the distance from the point (1, 2, 3) (a) XZ plane to the (b) Z axis

A: In this question we will use basic of geometry i.e concept of 3D geometry. We will first plot the…

Q: Find an equation for the plane in R® spanned by the points (1,5, 1) and (1, 4, 2).

A: We have to find the plane equation for two given points

Q: What will be the new coordinates of point M (-5, -7) after a rotation of 270° counterclockwise about…

A: To fins the new coordinate of point M (-5,-7) after a rotation of 270° anticlockwise about the…

Q: Determine the angle of rotation necessary to transform the equation in x and y into an equation in X…

A: Determine the angle of rotation necessary to transform the equation in x and y into an equation in X…

Q: Which of the following point is the projection of the point (0, 4, 2) on the yz-plane?

A: In this question we have to find the projection of the point on the yz-plane.

Q: Find b such that B = (0, b) has hyperbolic distance of u from the point A (0,0).

A: Given: B=(0,b) has  hyperbolic distance of u from the point A=(0,0) Formula used:

Q: Prove that three points (x1, yl), (x2, y2) and (x3, y3) are collinear ( on the same straight line)…

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Q: find the volu me of the o bócct Caused by the rotation oftne bouncled ared be twen y= aQ ,y=X…

A: Integrals and derivatives are considered as the inverses of each other. For example if we have a…

Q: (5) Let 8. u = 6. --(:)- and y = 1 What is the orthogonal projection of y to the line through u and…

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Q: The distance between the centers of the following two spheres: x2+56x+y2−54y+z2=−1512,…

A: A detailed solution is given below

Q: AJKL has vertices J(1,-3), K(4, –3), and L(3,1). Reflect AJKL across the y-axis and then rotate it…

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Q: 2 Let w = z². Find the image of the first quadrant in z-plane

A: