Question

Let the line \( L \) pass through the point \( (-3, 5, 2) \) and make equal angles with the positive coordinate axes. If the distance of \( L \) from the point \( (-2, r, 1) \) is \( \sqrt{\frac{14}{3}} \), then the sum of all possible values of \( r \) is :

• A. 16
• B. 12
• C. 10
• D. 6

Hint:

For a line making equal angles with axes, the direction ratios are always \( 1 : 1 : 1 \).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
A line making equal angles with axes has direction cosines \( (\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}) \). We use the formula for the distance of a point from a line in 3D.

Step 2: Detailed Explanation:
Direction of line \( L \): \( \vec{d} = (1, 1, 1) \).
Point on line \( L \): \( A = (-3, 5, 2) \).
Target point: \( P = (-2, r, 1) \).
Vector \( \vec{AP} = (-2 - (-3), r - 5, 1 - 2) = (1, r-5, -1) \).
The squared distance is given by:
\[ \text{dist}^2 = |\vec{AP}|^2 - \frac{(\vec{AP} \cdot \vec{d})^2}{|\vec{d}|^2} \]
\[ \frac{14}{3} = [1^2 + (r-5)^2 + (-1)^2] - \frac{(1 + r - 5 - 1)^2}{3} \]
\[ \frac{14}{3} = (r-5)^2 + 2 - \frac{(r-5)^2}{3} = \frac{2}{3}(r-5)^2 + 2 \]
\[ \frac{8}{3} = \frac{2}{3}(r-5)^2 \implies (r-5)^2 = 4 \]
So, \( r - 5 = \pm 2 \), which gives \( r = 7 \) or \( r = 3 \).
Sum of values = \( 7 + 3 = 10 \).

Step 3: Final Answer:
The sum of possible values of \( r \) is 10.