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 : $\left|\begin{array}{ccc}x-2& y-1& z+1\\ 3& -2& 1\\ 4& -3& 5\end{array}\right|=0$
$\Rightarrow (x-2)(-10+3)-(y-1)(15-4)+(z+1)(-1)=0$
$\Rightarrow -7x+14-11y+11-z-1=0$
$\Rightarrow 7x+11y+z=24$
$\therefore \alpha =7,\beta =11,\gamma =1$
$\alpha +\beta +\gamma =19$