JEE Advanced 2020 Paper 2Question 1
Let $\alpha,\beta,\gamma,\delta$ be real numbers such that $\alpha^2+\beta^2+\gamma^2\neq0$ and $\alpha+\gamma=1$. Suppose the point $(3,2,-1)$ is the mirror image of the point $(1,0,-1)$ with respect to the plane $\alpha x+\beta y+\gamma z=\delta$. Then which of the following statements is/are TRUE?
3D Geometry
Solution
Normal $(\alpha,\beta,\gamma)\parallel(3,2,-1)-(1,0,-1)=(2,2,0)$, so $\gamma=0$, and $\alpha+\gamma=1$ gives $\alpha=\beta=1$. Midpoint $(2,1,-1)$ lies on the plane: $\delta=2+1=3$. Then $\alpha+\beta=2$, $\delta-\gamma=3$, $\delta+\beta=4$ hold; $\alpha+\beta+\gamma=2\ne3$.