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ABC is a triangle in a plane with vertices A (2, 3, 5), B = (-1, 3, 2) and $C (λ, 5, μ)$ . If the median through ‘A’ is equally Inclined to the co-ordinate Axes then the value of $λ^{3} + μ^{3} + 5 =____$

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answer is 1348.

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Detailed Solution

D is midpoint of $\bar{B C}$

i.e $D = \frac{B + C}{2} = (\frac{λ - 1}{2}, 4, \frac{μ + 2}{2})$

Dr’s of $\bar{A D}$ are $(\frac{λ - 5}{2}, 1, \frac{μ - 8}{2})$

Median is making equal angles with coordinate axes:

$\frac{λ - 5}{2} = 1 = \frac{μ - 8}{2}$

$\Rightarrow λ = 7, μ = 10$

$\to λ^{3} + μ^{3} + 5 = 1348$

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