Question

Let Q and R be the feet of perpendiculars from the point P(a, a, a) on the lines x = y, z = 1 and x = –y, z = –1 respectively. If ∠QPR is a right angle, then 12a2 is equal to _____

Answer 1

The coordinates of $Q$ are given by:

$x = y, \quad z = 1 \quad \Rightarrow \quad Q(r, r, 1)$

The coordinates of $R$ are given by:

$x = -y, \quad z = -1 \quad \Rightarrow \quad R(k, -k, -1)$

Calculate vector $\overrightarrow{PQ}$:

$\overrightarrow{PQ} = (a - r)\hat{i} + (a - r)\hat{j} + (a - 1)\hat{k}$

Similarly, calculate vector $\overrightarrow{PR}$:

$\overrightarrow{PR} = (a - k)\hat{i} + (a + k)\hat{j} + (a + 1)\hat{k}$

Since $\overrightarrow{PQ} \perp \overrightarrow{PR}$:

$(a - r)(a - k) + (a - r)(a + k) + (a - 1)(a + 1) = 0$

Simplifying:

$a = 1 \quad \text{or} \quad -1$

Hence:

$12a^2 = 12$