Let ABC be a triangle with vertices at points A(2, 3, 5), B(-1, 3, 2) and C($$\lambda$$, 5, $$\mu$$) in three dimensional space. If the median through A is equally inclined with the axes, then $$(\lambda, \mu)$$ is equal to:

The median through vertex A is the line segment joining A to the midpoint of the opposite side BC. The vertices are given as A(2, 3, 5), B(-1, 3, 2), and C(λ, 5, μ). The midpoint M of BC is calculated by averaging the coordinates of B and C. The coordinates of M are: $$ M_x = \frac{-1 + \lambda}{2} $$ $$ M_y = \frac{3 + 5}{2} = \frac{8}{2} = 4 $$ $$ M_z = \frac{2 + \mu}{2} $$ So, M is at $$ \left( \frac{\lambda - 1}{2}, 4, \frac{\mu + 2}{2} \right) $$. The median is the line from A(2, 3, 5) to M. The direction ratios of AM are found by subtracting the coordinates of A from M: $$ a = \frac{\lambda - 1}{2} - 2 = \frac{\lambda - 1 - 4}{2} = \frac{\lambda - 5}{2} $$ $$ b = 4 - 3 = 1 $$ $$ c = \frac{\mu + 2}{2} - 5 = \frac{\mu + 2 - 10}{2} = \frac{\mu - 8}{2} $$ Thus, the direction ratios are $$ \left( \frac{\lambda - 5}{2}, 1, \frac{\mu - 8}{2} \right) $$. For the median to be equally inclined to the axes, the absolute values of the direction ratios must be equal, and since the angles are acute (as is standard for "equally inclined"), the direction ratios must all be positive. Therefore: $$ \left| \frac{\lambda - 5}{2} \right| = |1| = \left| \frac{\mu - 8}{2} \right| $$ Since |1| = 1, this implies: $$ \left| \frac{\lambda - 5}{2} \right| = 1 \quad \text{and} \quad \left| \frac{\mu - 8}{2} \right| = 1 $$ Solving these equations: $$ \left| \lambda - 5 \right| = 2 \quad \text{and} \quad \left| \mu - 8 \right| = 2 $$ This gives two cases for each equation: For λ: $$ \lambda - 5 = 2 \quad \Rightarrow \quad \lambda = 7 $$ $$ \lambda - 5 = -2 \quad \Rightarrow \quad \lambda = 3 $$ For μ: $$ \mu - 8 = 2 \quad \Rightarrow \quad \mu = 10 $$ $$ \mu - 8 = -2 \quad \Rightarrow \quad \mu = 6 $$ The possible pairs (λ, μ) are (7, 10), (7, 6), (3, 10), and (3, 6). However, for the direction ratios to be positive (ensuring acute angles): $$ \frac{\lambda - 5}{2} > 0 \quad \Rightarrow \quad \lambda > 5 $$ $$ \frac{\mu - 8}{2} > 0 \quad \Rightarrow \quad \mu > 8 $$ Applying these inequalities: - λ = 7 > 5 is valid, but λ = 3 < 5 is invalid. - μ = 10 > 8 is valid, but μ = 6 < 8 is invalid. Thus, the only valid pair is (λ, μ) = (7, 10). Verifying with the direction ratios: For λ = 7, μ = 10: $$ a = \frac{7 - 5}{2} = \frac{2}{2} = 1 > 0 $$ $$ b = 1 > 0 $$ $$ c = \frac{10 - 8}{2} = \frac{2}{2} = 1 > 0 $$ The direction ratios are (1, 1, 1), with equal absolute values and all positive, confirming the median is equally inclined to the axes. Comparing with the options: - A. (10, 7) - B. (7, 5) - C. (7, 10) - D. (5, 7) The pair (7, 10) corresponds to option C. Hence, the correct answer is Option C.