Let `A(2, 3, 5), B(-1, 3, 2), C(lambda, 5, mu)` are the vertices of a triangle and its median through A(I.e.,) AD is equally inclined to the coordinates axes.
Q. On the basis of the above information answer the following
Q. The value of `2lambda-mu` is equal to

Options

A. `13`
B. `4`
C. `3`
D. None of these

Answer

To solve the problem, we will follow these steps:

### Step 1: Identify the coordinates of points A, B, and C
Given the points:
- A(2, 3, 5)
- B(-1, 3, 2)
- C(λ, 5, μ)

### Step 2: Find the midpoint D of line segment BC
The coordinates of the midpoint D 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 B and C:
\[ D = \left( \frac{-1 + λ}{2}, \frac{3 + 5}{2}, \frac{2 + μ}{2} \right) = \left( \frac{λ - 1}{2}, 4, \frac{μ + 2}{2} \right) \]

### Step 3: Determine the direction ratios of line AD
The direction ratios of line AD can be found using the coordinates of points A and D:
\[ \text{Direction Ratios} = (x_D - x_A, y_D - y_A, z_D - z_A) \]
Substituting the coordinates of A and D:
\[ \text{Direction Ratios} = \left( \frac{λ - 1}{2} - 2, 4 - 3, \frac{μ + 2}{2} - 5 \right) \]
This simplifies to:
\[ \text{Direction Ratios} = \left( \frac{λ - 1 - 4}{2}, 1, \frac{μ + 2 - 10}{2} \right) = \left( \frac{λ - 5}{2}, 1, \frac{μ - 8}{2} \right) \]

### Step 4: Set the direction ratios equal
Since the median AD is equally inclined to the coordinate axes, the direction ratios must be equal:
\[ \frac{λ - 5}{2} = 1 = \frac{μ - 8}{2} \]

### Step 5: Solve for λ and μ
From the first equation:
\[ \frac{λ - 5}{2} = 1 \implies λ - 5 = 2 \implies λ = 7 \]
From the second equation:
\[ \frac{μ - 8}{2} = 1 \implies μ - 8 = 2 \implies μ = 10 \]

### Step 6: Calculate the value of 2λ - μ
Now we can substitute the values of λ and μ into the expression:
\[ 2λ - μ = 2(7) - 10 = 14 - 10 = 4 \]

### Final Answer
The value of \(2λ - μ\) is equal to **4**.

---