3D Geometry
Section Formula
Grade 12
Question:
<p>Let \(A(2, 3, 5)\), \(B(-1, 3, 2)\) and \(C(\lambda, 5, \mu)\) be the vertices of a \(\triangle ABC\). If the median through \(A\) is equally inclined to the coordinate axes, then</p>
<p>\(5\lambda - 8\mu = 0\)</p>
<p>\(8\lambda - 5\mu = 0\)</p>
<p>\(10\lambda - 7\mu = 0\)</p>
<p>\(7\lambda - 10\mu = 0\)</p>
Step-by-Step Solution
Key Concept: The median from A passes through the midpoint of BC. For a line to be equally inclined to all three coordinate axes, the direction ratios must be proportional to (1, 1, 1), meaning equal direction cosines in magnitude.
Step 1: Find the midpoint M of BC. Midpoint M = $\left(\frac{-1+\lambda}{2}, \frac{3+5}{2}, \frac{2+\mu}{2}\right) = \left(\frac{\lambda-1}{2}, 4, \frac{\mu+2}{2}\right)$ Step 2: Find direction ratios of median AM. Direction ratios of AM = $\left(\frac{\lambda-1}{2}-2, 4-3, \frac{\mu+2}{2}-5\right)$ = $\left(\frac{\lambda-5}{2}, 1, \frac{\mu-8}{2}\right)$ Multiply by 2: $(\lambda-5, 2, \mu-8)$ Step 3: Apply condition of equal inclination to coordinate axes. For equal inclination to coordinate axes, direction ratios must be proportional to (1, 1, 1). Therefore: $\lambda - 5 = 2 = \mu - 8$ Step 4: Solve for λ and μ. From $\lambda - 5 = 2$: $\lambda = 7$ From $2 = \mu - 8$: $\mu = 10$ ∴ Answer: A ($\lambda = 7, \mu = 10$)
Correct Answer: A