ABC is a triangle in a plane with vertices `A(2,3,5),B(-1,3,2)` and `C(lamda,5,mu)`. If the median through A is equally inclined to the coordinate axes, then the value of `lamda^(3)+mu^(3)+5` is

Options

A. 676
B. 1130
C. 1348
D. 1077

Explanation

To solve the problem, we need to find the value of \( \lambda^3 + \mu^3 + 5 \) given that the median through point A is equally inclined to the coordinate axes.

### Step-by-Step Solution:

1. **Identify the Points**:
We have the vertices of triangle ABC as:
- \( A(2, 3, 5) \)
- \( B(-1, 3, 2) \)
- \( C(\lambda, 5, \mu) \)

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

3. **Find the Direction Ratios of AM**:
The direction ratios of line segment \( AM \) can be found by subtracting the coordinates of A from M:
\[ AM = \left( \frac{-1 + \lambda}{2} - 2, 4 - 3, \frac{2 + \mu}{2} - 5 \right) \]
Simplifying this gives:
\[ AM = \left( \frac{\lambda - 5}{2}, 1, \frac{\mu - 8}{2} \right) \]

4. **Condition for Equal Inclination**:
Since the median \( AM \) is equally inclined to the coordinate axes, the direction ratios must be proportional to \( (1, 1, 1) \). Therefore, we can set:
\[ \frac{\lambda - 5}{2} = k, \quad 1 = k, \quad \frac{\mu - 8}{2} = k \]
where \( k \) is some constant.

5. **Solve for \( \lambda \) and \( \mu \)**:
From \( k = 1 \):
- For \( \lambda \):
\[ \frac{\lambda - 5}{2} = 1 \implies \lambda - 5 = 2 \implies \lambda = 7 \]
- For \( \mu \):
\[ \frac{\mu - 8}{2} = 1 \implies \mu - 8 = 2 \implies \mu = 10 \]

6. **Calculate \( \lambda^3 + \mu^3 + 5 \)**:
Now we can calculate:
\[ \lambda^3 = 7^3 = 343, \quad \mu^3 = 10^3 = 1000 \]
Therefore:
\[ \lambda^3 + \mu^3 + 5 = 343 + 1000 + 5 = 1348 \]

### Final Answer:
The value of \( \lambda^3 + \mu^3 + 5 \) is \( \boxed{1348} \).