JEE Advanced 2020 — Paper 2 — Question 43

The mirror image implies that the given points are symmetric with respect to the plane.

Hence the line joining the points is perpendicular to the plane, so its direction ratios are proportional to the normal vector $(\alpha, \beta, \gamma)$ of the plane. Direction ratios of the line joining $(3,2,-1)$ and $(1,0,-1)$ are $(2,2,0)$. Since $(2,2,0) \propto (\alpha, \beta, \gamma)$, we get $\frac{\alpha}{2} = \frac{\beta}{2} = \frac{\gamma}{0}$, which gives $\gamma = 0$ and $\alpha = \beta$. Given $\alpha + \gamma = 1$ $\Rightarrow \alpha = 1$, and thus $\beta = 1$. So the plane equation is $x + y = \delta$. The midpoint of the two points lies on the plane. Midpoint $M = \left(\frac{3+1}{2}, \frac{2+0}{2}, \frac{-1-1}{2}\right) = (2,1,-1)$. Substitute $M$ into the plane: $2 + 1 = \delta \Rightarrow \delta = 3$. Now verify options: A) $\alpha+\beta=1+1=2$, true;

B) $\delta-\gamma=3-0=3$, true;

C) $\delta+\beta=3+1=4$, true;

D) $\alpha+\beta+\gamma=2$, $\delta=3$,

false. Hence A, B, C are correct.