Question

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:

• A. \(5\lambda-8\mu=0\)
• B. \(8\lambda-5\mu=0\)
• C. \(10\lambda-7\mu=0\)
• D. \(7\lambda-10\mu=0\)

Hint:

Find the midpoint D of BC, then the direction ratios of AD; equal inclination means all direction ratios have equal magnitude.

Answer 1

The Correct Option is C

Midpoint of BC: \(D = \left(\frac{\lambda-1}{2},\,4,\,\frac{\mu+2}{2}\right)\).
Direction ratios of AD: \(\left(\frac{\lambda-5}{2},\,1,\,\frac{\mu-8}{2}\right)\).
Equally inclined to the axes means \(l^2 = m^2 = n^2\), so \(\left|\frac{\lambda-5}{2}\right| = 1 = \left|\frac{\mu-8}{2}\right|\).
Thus \(\lambda = 7\) or \(3\) and \(\mu = 10\) or \(6\).
Checking the options, \(10\lambda - 7\mu = 0\) holds for \(\lambda = 7,\ \mu = 10\) (70 - 70 = 0); no other option is satisfied by any allowed pair.
Answer: (3) \(10\lambda-7\mu=0\)