Question:
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:
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:
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Median from A goes to the midpoint of BC; equal inclination means all direction ratios are equal.
Updated On: Nov 4, 2026
- \((10,7)\)
- \((7,5)\)
- \((7,10)\)
- \((5,7)\)
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The Correct Option is C
Solution and Explanation
Midpoint of BC: \(D=\left(\frac{\lambda-1}{2},\,4,\,\frac{\mu+2}{2}\right)\).
Direction ratios of AD: \(\left(\frac{\lambda-1}{2}-2,\ 4-3,\ \frac{\mu+2}{2}-5\right)=\left(\frac{\lambda-5}{2},\ 1,\ \frac{\mu-8}{2}\right)\).
Equally inclined with the axes means the direction ratios are equal: \(\frac{\lambda-5}{2}=1=\frac{\mu-8}{2}\).
\(\lambda=7,\ \mu=10\).
Answer: (3) \((7,10)\)
Direction ratios of AD: \(\left(\frac{\lambda-1}{2}-2,\ 4-3,\ \frac{\mu+2}{2}-5\right)=\left(\frac{\lambda-5}{2},\ 1,\ \frac{\mu-8}{2}\right)\).
Equally inclined with the axes means the direction ratios are equal: \(\frac{\lambda-5}{2}=1=\frac{\mu-8}{2}\).
\(\lambda=7,\ \mu=10\).
Answer: (3) \((7,10)\)
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