A triangle has vertices (1, 1), (3, 7) and (5, 4). What is the distance between the centroid and the line 2x = -y - 2? Hint: The coordinate of the centroid can be obtained from the average of the x and y-coordinates of the vertices. 2. Find the equation of a line passing through (4, -5) when the intercepts are numerically equal but of opposite signs. 3. Find the point (x,4) which is 5 units from the line 12x + 5y - 6 = 0

Q: Find the equation of the plane with the given description. Passes through (6, 1, 5) and is parallel…

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Q: Q2: Find the equation of the plane that pass through the point P(-1,4,2) and containing the line of…

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Q: 4] Find the equation of the plane that contains the point (a, 0, 0), (0, b, 0) and (0, 0, c).

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Q: 10) Find the equation of the line of the intersection of the planes 2x+3y+7z+2=D0, x+2y-3z+5=D0.

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Q: Determine the intersection of 3x – 4y – 5z = 0 with [x, y, z] = [3, 1, 1] + t [2, -1, 2].

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Q: Find the equation of the plane that passes through the points 4 = (2,4,-3), B = (3,-1,3) and y - 4 -…

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Q: 2. Find the distance between the given parallel planes. 2x +y+ 2z = 8 and 4x +2y + 4z +5 = 0

A: # we are entitled to solve one question at a time, please resubmit the other questions if you wish…

Q: 6. Find the equation of the plane that passes through (3,1,2) and (-1,2,–1) and is parallel to the…

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Q: Find the distance from the point (0, -1, -2) to the plane - 4y – 4z = 5.

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Q: Find the point on the plane x-y+z=7 that is closest to the point (2,4,6).

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Q: 1. Find the distance from the point S(2,-3,4) to the plane x+2y+ 2z =13

A: "Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Find the line of intersection of the given planes. 4x + y-z = 0 and 2x - y + 3z 4 %3D

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Q: Which of the given points are the points on the hyperboloid x2-y?+4z2 = 4 where the normal line is…

A: Solution is in step 2

Q: The distance from the point S(7, 7, -6) to the plane 2x + 11y + 10z = 9

A: option C 2215 is correct detailed solution is given below

Q: The plane that passes through  (9, 0, −3)  and contains the line  x = 6 − 3t, y = 2 + 5t, z = 4 +…

A: Given : A point (9,0,-3) where the plane passes through.             A system of equations for line…

Q: Find the linear equation of the plane through the point (2,6,9) and parallel to the plane…

A: Equation of plane parallel to  x+5y+4z+4=0 is given by,x+5y+4z+4+λ=0

Q: Find a plane containing the point (-3,-3,5) and the line x(t) y(t) z(t) = = 8 + 4t -4-6t -7-3t

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Q: Find an equation of the plane that contains the point (2, 0, 3) and the line x =-1+t, y =t, z=-4+21…

A: It is given that the required plane contains the point 2,0,3 and the line x=-1+t,y=t,z=-4+2t. The…

Q: Consider a circle C in the xy-plane. If it is given that the points (2, 3),(2, −1) and (0, 3) are on…

A: Given points are (2, 3),(2, −1) and (0, 3)

Q: Find equations of the line through the point (1, −1, 1) perpendicular to the line 3x = 2y = z and…

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Q: Find the line of intersection of the given planes. 4x + y + z = 0 and 2x -y + 3z = 2

A: Given planes are: 4x + y + z = 0 2x - y + 3z = 0 Let us consider, z = t Then we get, 4x + y = -t 2x…

Q: The plane through the points (2, 1, –1), (0, –2, 0), and (1, – 1, 2) is parallel to the line x=1-t,…

A: Equation of plane passing through x1, y1, x2, y2, x3, y3 is given by,…

Q: The equation of the line tangent to the intersection of two surfaces 2 = a2 and z 25- y? at the…

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Q: An equation for the level surface passing through the point (2,0, - 1) is z =

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Q: Find an equation of the plane through (0, –9, –1) that is perpendicular to the line x – 2 = 3t, y+3…

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Q: A Find the Distance from the point (2,-6) to the line 3x+ 4y t6=0 B Find the distance between the…

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Q: 2. Write the symmetric equation of a line through the points (0, 2,8) and (3,-3, 4). Find where this…

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Q: Find the point on the plane 2x + y + 4z = 5 that is closest to the point P(4, 3, 5). (x, y, z) =…

A: We are given a plane 2x + y +4z = 5, and we are asked to find the co-ordinates of the point on the…

Q: A point moves in the (x, y) plane on the line 2x+ 3y −4 = 0. Where will it be when the sum of the…

A: We have to find a point from where the sum of square of its distances from 1,0 and -1,0 is smallest.

Q: 4. Find an equation for the plane in R³ spanned by the points (1,5, 1) and (1, 4, 2).

A: The two vectors are given as v1 = 151v2 = 142 Let xyz∈span{v1 , v2} Then xyz = mv1+nv2= m151+n142=…

Q: Solve for the linear equation of the plane through the point (1,9,10)(1,9,10) and parallel to the…

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Q: Determine the x coordinate of the point on the plane Π:−3x−2y+2z=2 closest to the point (2,5,−5).

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Q: A quadrilateral has the following vertices given:  P1 (1, 4), P2 (3, 4), P3 (3, 6), P4 (1, 6).  The…

A: Given points are :  P1(1,4) , P2(3,4) , P3(3,6) , P4(1,6). We will first plot those points in a…

Q: Find the shortest distance between (3,4,1 ) and the plane X+y +Z = -2

A: Using distance between plane and point formula we can calculate.

Q: 2. Find the coordinates of the y-intercept of the plane (x, y, z)=(4,6,5) +s(2,3,1) + t(3,6,12)

A: To find out the coordinate of y-intercept.

Q: Find the equation of the plane whose has the line x=2-3t, y=3 + t, z = 5t and passes through the…

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Q: Find the minimum distance from the point to the plane x - y + z = 9. (Hint: To simplify the…

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Q: Find the line through (1,-1,1)and normal to the line  3x=2y=z , and parallel to the plance  x+y-z=0…

A: Equation of the line: The equation of a line passing through the point (x0, y0, z0) is given by (x,…

Q: Find the minimum distance from the point (8, 3, 4) to the plane x - y + z = 4. (Hint: To simplify…

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Q: 9. The line through (0,-7, 0) perpendicular to the plane x + 2y + 2z = 13

A: Given point is (0,-7,0) and plane equation is x+2y+2z=13 here we have to find the parametric…

Q: A normal equation of the plane that contains the line 1₁: (x, y, z) = (1, -1, 2) + t (9, 1, 3), t E…

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Q: what is the equation of the plane perpendicular to the intersection line of the x-2y+2z= 3  and…

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Q: 4] Find the equation of the plane that contains the point (a, 0, 0), (0, b, 0) and (0,0, c).

A: As per our company guidelines we are supposed to answer only one question kindly post other parts in…

Q: Find an equation of the plane through (4, 1, –4) that is perpendicular to the line x – 2 = 5t, y+3 =…

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Q: 4] Find the equation of the plane that contains the point (а, 0, 0), (0, b, 0) and (0, 0, c).

A: First find two vector and then take cross product which is normal vector  Now use equation of plane

Q: Find the equation of the line perpendicular to the plane x−2y +z = 4 passing through the point (1, 0

A: We have to find equation

Q: Find the minimum distance from the point to the surface z = √4 - 2x - 2y. (Hint: To simplify the…

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Q: Find an equation of the plane through (-4, 3, –8) that is perpendicular to the line x – 2 = t, y +3…

A: Given line : x-2=t , y+3=2t ,  z=-tx-21=t , y+32=t , z-1=tx-21= y+32= z-0-1 comparing with standard…