Let the plane 2x + 3y + z + 20 = 0 be rotated through a right angle ab

1.	Let the plane 2x + 3y + z + 20 = 0 be rotated through a right angle about its line of intersection with the plane x – 3y + 5z = 8. If the mirror image of the point (2, -1/2, 2) in the rotated plane is B(a, b, c), then :(A) a/8 = b/5 = c/-4(B) a/4 = b/5 = c/-2(C) a/8 = b/-5 = c/4(D) a/4 = b/5 = c/2
Answer» Correct option is (A) a/8 = b/5 = c/-4 Let equation of rotated plane be : (2x + 3y + z + 20) + λ (x – 3y + 5z – 8) = 0 (2 + λ )x + (3 – 3λ )y + (1 + 5l)z + 20 – 8λ = 0 Above plane is perpendicular to 2x + 3y + z + 20 = 0 So, (2 + λ ).2 + (3 – 3λ ).3 + (1 + 5λ).1 = 0 ⇒ λ = 7 ⇒ Equation of rotated plane : x – 2y + 4z – 4 = 0 Mirror image of A(2, -1/2, 2) in rotated plane is B(a, b, c) Equation of AB : \(\frac{x-2}1=\frac{(y+1)/2}{-2}=\frac{z-2}4=k\) Let coordinate of B be (2 + k/2, -1/2 - k, 2 + 2k) which will lie on the plane x - 2y + 4z - 4 = 0 Hence k = -2/3 Therefore B is (4/3, 5/6, -2/3) = (8/6, 5/6, -4/6) So, a/8, b/5 = c/-4

Let the plane 2x + 3y + z + 20 = 0 be rotated through a right angle about its line of intersection with the plane x – 3y + 5z = 8. If the mirror image of the point (2, -1/2, 2) in the rotated plane is B(a, b, c), then :(A) a/8 = b/5 = c/-4(B) a/4 = b/5 = c/-2(C) a/8 = b/-5 = c/4(D) a/4 = b/5 = c/2

Answer»

Correct option is (A) a/8 = b/5 = c/-4

Let equation of rotated plane be :

(2x + 3y + z + 20) + λ (x – 3y + 5z – 8) = 0

(2 + λ )x + (3 – 3λ )y + (1 + 5l)z + 20 – 8λ = 0

Above plane is perpendicular to 2x + 3y + z + 20 = 0

So, (2 + λ ).2 + (3 – 3λ ).3 + (1 + 5λ).1 = 0

⇒ λ = 7

⇒ Equation of rotated plane : x – 2y + 4z – 4 = 0

Mirror image of A(2, -1/2, 2) in rotated plane is B(a, b, c)

Equation of AB : \(\frac{x-2}1=\frac{(y+1)/2}{-2}=\frac{z-2}4=k\)

Let coordinate of B be (2 + k/2, -1/2 - k, 2 + 2k) which will lie on the plane x - 2y + 4z - 4 = 0

Hence k = -2/3

Therefore B is (4/3, 5/6, -2/3) = (8/6, 5/6, -4/6)

So, a/8, b/5 = c/-4