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:

\[ 12a^2 = 12 \]

Answer 2

To solve the problem, we determine the coordinates of points Q and R, verify that ∠QPR is a right angle, and compute \( 12a^2 \) to confirm it matches the given range \([12, 12]\). 

Step 1: Finding Coordinates of Q and R

The line \( x = y, z = 1 \) can be written in parametric form as:

\[ (x, y, z) = (t, t, 1) \]

The perpendicular from \( P(a, a, a) \) to this line satisfies:

\[ (t - a)(1, 1, 1) \cdot (1, 1, 0) = 0 \] \[ 2(t - a) = 0 \Rightarrow t = a \]

Hence, \( Q = (a, a, 1) \).

For the line \( x = -y, z = -1 \), the parametric form is:

\[ (x, y, z) = (s, -s, -1) \]

The perpendicular from \( P(a, a, a) \) to this line satisfies:

\[ (s - a, -s - a, -1 - a) \cdot (1, -1, 0) = 0 \] \[ (s - a) - (-s - a) = 0 \Rightarrow s = 0 \]

Thus, \( R = (0, 0, -1) \).

Step 2: Verifying ∠QPR = 90°

\[ \overrightarrow{PQ} = Q - P = (0, 0, 1 - a), \quad \overrightarrow{PR} = R - P = (-a, -a, -1 - a) \]

Dot product:

\[ \overrightarrow{PQ} \cdot \overrightarrow{PR} = 0 \cdot (-a) + 0 \cdot (-a) + (1 - a)(-1 - a) \] \[ (1 - a)(-1 - a) = 0 \Rightarrow a^2 + 2a + 1 = 0 \Rightarrow (a + 1)^2 = 0 \] \[ \therefore a = -1 \]

Step 3: Calculating \( 12a^2 \)

\[ 12a^2 = 12(-1)^2 = 12 \]

Hence, the value lies within the range \([12, 12]\).

Conclusion: \( 12a^2 = 12 \), confirming the required condition.