Question

If the equation of plane passing through the mirror image of a point $(2,3,1)$ with respect to line $\frac{x+1}{2}=\frac{y-3}{1}=\frac{z+2}{-1}$ and containing the line $\frac{x-2}{3}=\frac{1-y}{2}=\frac{z+1}{1}$ is $\alpha x+\beta y+\gamma z=24$, hen $\alpha+\beta+\gamma$ is equal to :

• A. 20
• B. 19
• C. 18
• D. 21

Answer 1

Answer: (B)

Line $\frac{x+1}{2}=\frac{y-3}{1}=\frac{z+2}{-1}$

$ \overrightarrow{ PM }=(2 \lambda-3, \lambda,-\lambda-3)$
$\overrightarrow{ PM } \perp(2 \hat{ i }+\hat{ j }-\hat{ k }) $
$4 \lambda-6+\lambda+\lambda+3=0 \Rightarrow \lambda=\frac{1}{2} $
$\therefore M \equiv\left(0, \frac{7}{2}, \frac{-5}{2}\right) $
$\therefore \text { Reflection }(-2,4,-6)$
Plane : $\begin{vmatrix} x -2 & y -1 & z +1 \\ 3 & -2 & 1 \\ 4 & -3 & 5\end{vmatrix}=0 $
$ \Rightarrow ( x -2)(-10+3)-( y -1)(15-4)+( z +1)(-1)=0 $
$\Rightarrow -7 x +14-11 y +11- z -1=0 $
$\Rightarrow 7 x +11 y + z =24 $
$ \therefore \alpha =7, \beta=11, \gamma=1 $
$\alpha+\beta+\gamma=19$