Let A (2, 3, 5), B (-1, 3, 2) and C(λ, 5, μ) be the vertices of a ΔABC. If the median through A is equally inclined to the coordinate axes, then :

Midpoint of B and C is (λ-1/2, 4, μ+2/2)
Now, D.R.'s of median will be (λ-1/2 -2, 4-3, μ+2/2 -5) = (λ-5/2, 1, μ-8/2)
(λ-5/2, 1, μ-4)
D.C.'s will be same since it is equally inclined to the coordinate axis.
λ-5/2 = 1 = μ-4
λ = 7/2 and μ = 5
This is incorrect, let's recheck
Midpoint of B and C = ((λ - 1)/2, 4, (μ + 2)/2)
Direction ratios of the median through A are: ((λ - 1)/2 - 2, 4 - 3, (μ + 2)/2 - 5) = ((λ - 5)/2, 1, (μ - 8)/2)
Since the median is equally inclined to the coordinate axes, the direction ratios are equal:
(λ - 5)/2 = 1
λ - 5 = 2
λ = 7
(μ - 8)/2 = 1
μ - 8 = 2
μ = 10
Therefore, 7λ - μ = 7(7) - 10 = 49 - 10 = 39 ≠ 0
Let's assume the direction cosines are equal:
(λ - 5)/2 = 1 = (μ - 8)/2
λ = 7, μ = 10
Then 7λ - μ = 7(7) - 10 = 39 ≠ 0
10λ - μ = 10(7) - 10 = 60 ≠ 0
5λ - μ = 5(7) - 10 = 25 ≠ 0
8λ - μ = 8(7) - 10 = 46 ≠ 0
If the median is equally inclined to the axes, then the direction ratios are equal:
(λ - 5)/2 = 1 = (μ - 8)/2
λ - 5 = 2 => λ = 7
μ - 8 = 2 => μ = 10
7λ - μ = 7(7) - 10 = 39
Therefore, none of the options are correct.