Question

Let \( A(2, 3, 5), B(-1, 3, 2), C(\lambda, 5, \mu) \) be the vertices of \( \triangle ABC \). If the median through the vertex \( A \) is equally inclined to the coordinate axes, then

• A. \( 5\lambda - 8\mu = 0 \)
• B. \( 8\lambda - 5\mu = 0 \)
• C. \( 10\lambda - 7\mu = 0 \)
• D. \( 7\lambda - 10\mu = 0 \)

Hint:

When a vector is equally inclined to all coordinate axes, its direction ratios are equal in magnitude.

Answer 1

The Correct Option is C

Step 1: The median from vertex \( A \) goes to the midpoint \( M \) of side \( BC \). First find midpoint \( M \): \[ M = \left( \frac{-1 + \lambda}{2}, \frac{3 + 5}{2}, \frac{2 + \mu}{2} \right) = \left( \frac{\lambda - 1}{2}, 4, \frac{\mu + 2}{2} \right) \] Step 2: The direction vector of the median from \( A(2, 3, 5) \) to \( M \) is: \[ \vec{AM} = \left( \frac{\lambda - 1}{2} - 2, 1, \frac{\mu + 2}{2} - 5 \right) = \left( \frac{\lambda - 5}{2}, 1, \frac{\mu - 8}{2} \right) \] Step 3: The median is equally inclined to all coordinate axes, so direction ratios are equal in magnitude: \[ \left| \frac{\lambda - 5}{2} \right| = \left| 1 \right| = \left| \frac{\mu - 8}{2} \right| \Rightarrow \frac{\lambda - 5}{2} = 1,
\frac{\mu - 8}{2} = \pm 1 \] Solving: \[ \lambda = 7,
\mu = 10 \text{ or } \mu = 6 \] Only one correct relation from given options satisfies the constraint for equal inclination. Plugging into options shows: \[ 10\lambda - 7\mu = 0
\text{is satisfied for } \lambda = 7, \mu = 10 \]