`A B C` is a triangle and A=(2,3,5),B=(-1,3,2) and C= `(lambda,5,mu).` If the median through `A` is equally inclined to the axes, then find the value of `(lambda , mu)`

Options

A. `(10,7)`
B. `(7,5)`
C. `(7,10)`
D. `(5,7)`

Answer

To solve the problem, we need to find the coordinates \( ( \lambda, \mu ) \) of point C such that the median through point A is equally inclined to the axes. Here are the steps to solve the problem:

### Step 1: Find the midpoint D of segment BC
The coordinates of points B and C are given as:
- \( B = (-1, 3, 2) \)
- \( C = (\lambda, 5, \mu) \)

The coordinates of the midpoint \( D \) of segment \( BC \) can be calculated using the midpoint formula:
\[ D = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2} \right) \]
Substituting the coordinates of points B and C:
\[ D = \left( \frac{-1 + \lambda}{2}, \frac{3 + 5}{2}, \frac{2 + \mu}{2} \right) = \left( \frac{\lambda - 1}{2}, 4, \frac{\mu + 2}{2} \right) \]

### Step 2: Find the direction ratios of line AD
The coordinates of point A are given as:
- \( A = (2, 3, 5) \)

The direction ratios of line \( AD \) can be found by subtracting the coordinates of A from D:
\[ \text{Direction ratios of } AD = \left( \frac{\lambda - 1}{2} - 2, 4 - 3, \frac{\mu + 2}{2} - 5 \right) \]
This simplifies to:
\[ \text{Direction ratios of } AD = \left( \frac{\lambda - 5}{2}, 1, \frac{\mu - 8}{2} \right) \]

### Step 3: Set the condition for equal inclination to the axes
For the median \( AD \) to be equally inclined to the axes, the absolute values of the direction ratios must be equal:
\[ \left| \frac{\lambda - 5}{2} \right| = |1| = \left| \frac{\mu - 8}{2} \right| \]

This gives us two equations:
1. \( \frac{\lambda - 5}{2} = 1 \)
2. \( \frac{\mu - 8}{2} = 1 \)

### Step 4: Solve the equations
**From the first equation:**
\[ \frac{\lambda - 5}{2} = 1 \implies \lambda - 5 = 2 \implies \lambda = 7 \]

**From the second equation:**
\[ \frac{\mu - 8}{2} = 1 \implies \mu - 8 = 2 \implies \mu = 10 \]

### Final Answer
Thus, the values of \( \lambda \) and \( \mu \) are:
\[ (\lambda, \mu) = (7, 10) \]