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

Correct Answer: 12

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:

\[ 12a^2 = 12 \]