Question
$$ABC$$ is a triangle where $$A = \left( {2,\,3,\,5} \right),\,B = \left( { - 1,\,3,\,2} \right)$$ and $$C = \left( {\lambda ,\,5,\,\mu } \right).$$ If the median through $$A$$ is equally inclined with the axes then :
A.
$$\lambda = 14,\,\mu = 20$$
B.
$$\lambda = 7,\,\mu = 10$$
C.
$$\lambda = \frac{7}{2},\,\mu = 5$$
D.
$$\lambda = 10,\,\mu = 7$$
Answer :
$$\lambda = 7,\,\mu = 10$$
Solution :
Centroid $$G = \left( {\frac{{2 - 1 + \lambda }}{3},\,\frac{{3 + 3 + 5}}{3},\,\frac{{5 + 2 + \mu }}{3}} \right).$$
Direction ratios of $$AG$$ are $$2 - \frac{{1 + \lambda }}{3},\,3 - \frac{{11}}{3},\,5 - \frac{{7 + \mu }}{3},{\text{ i}}{\text{.e}}{\text{., }}\frac{{5 - \lambda }}{3},\,\frac{{ - 2}}{3},\,\frac{{8 - \mu }}{3}.$$
As $$AG$$ is equally inclined with the axes, the direction ratios are $$1,\,1,\,1$$ also.
$$\therefore \frac{{\frac{{5 - \lambda }}{3}}}{1} = \frac{{\frac{{ - 2}}{3}}}{1} = \frac{{\frac{{8 - \mu }}{3}}}{1} \Rightarrow 5 - \lambda = - 2 = 8 - \mu $$