Question

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 
\((2,−\frac{1}{2},2) \)
in the rotated plane is B( a, b, c),then

• A. \(\frac{a}{8}=\frac{b}{5}=\frac{c}{-4}\)
• B. \(\frac{a}{4}=\frac{b}{-5}=\frac{c}{-2}\)
• C. \(\frac{a}{8}=\frac{b}{-5}=\frac{c}{4}\)
• D. \(\frac{a}{4}=\frac{b}{5}=\frac{c}{2}\)

Answer 1

The Correct Option is A

The correct answer is (A) : \(\frac{a}{8}=\frac{b}{5}=\frac{c}{-4}\)
Consider the equation of plane, 
P : (2x + 3y + z + 20) + λ(x – 3y + 5z – 8) = 0
P : (2 + λ)x + (3 – 3λ)y + (1 + 5λ)z + (20 – 8λ) = 0
∵ Plane P is perpendicular to 2x + 3y + z + 20 = 0
So, 4 + 2λ + 9 – 9λ + 1 + 5λ = 0 ,vso  λ=7
P :9x – 18y + 36z – 36 = 0
Or P :x – 2y + 4z = 4
If image of \((2,−\frac{1}{2},2)\)
 in plane P is (a, b, c) then
\(\frac{(a−2)}{1}=\frac{(b+\frac{1}{2})}{−2}=\frac{(c−2)}{4 }\)
and \(\frac{(a+2)}{2}−2(\frac{b−\frac{1}{2}}{2})+4(\frac{c+2}{2})=4\)
Clearly 
\(a=\frac{4}{3},b=\frac{5}{6} and \) \(c=−\frac{2}{3} \)
So, a :b : c = 8 : 5 : – 4