Question

Let P be a plane passing through the points $(2,1,0),(4,1,1)$ and $(5,0,1)$ and R be any point $(2,1,6)$. Then the image of R in the plane P is :

• A. $(6,5,2)$
• B. $(6,5,-2)$
• C. $(4,3,2)$
• D. $(3,4,-2)$

Answer 1

Answer: (B)

Plane passing through : $(2,1,0),(4,1,1)$ and $(5,0,1)$
$\left|\begin{array}{ccc}x-2& y-1& z\\ 2& 0& 1\\ 3& -1& 1\end{array}\right|=0$
$\Rightarrow x+y-2z=3$
Let I and F are respectively image and foot of perpendicular of point P in the plane.
$e{q}^n$ of line PI $\frac{x-2}{1}=\frac{y-1}{1}=\frac{z-6}{-2}=\lambda \left(say\right)$
Let I $\left(\lambda +2,\lambda +1,-2\lambda +6\right)$
$\Rightarrow F\left(2+\frac{\lambda }{2},1+\frac{\lambda }{2},-\lambda +6\right)$
F lies in the plane
$\Rightarrow 2+\frac{\lambda }{2}+\frac{\lambda }{2}+2\lambda -12-3=0$
$\Rightarrow \lambda =4$
$\Rightarrow I\left(6,5,-2\right)$