Question

Find the ratio in which the point $P(5,4,-6)$ divides the line segment joining the points $A(3,2,-4)$ and $B(9,8,-10)$. Also, find the harmonic conjugate of $P$.

Answer 1

Let the ratio be $k:1$

$\therefore 5=\frac{9k+3}{k+1}=5k+5=9k+3$

$\therefore 4k=2$

$\Rightarrow k=\frac{1}{2}$

$\therefore (5,4,-6)$ divides $A(3,2,-4)$ & $B(9,8,-10)$ in the ratio $1:2$ finally.

Let $Q(x,y,z)$ divide $A:B$ in the ratio $1:2$ externally.

$x=\frac{9-6}{-1}=-3$

$y=\frac{8-4}{-1}=-4$

$z=\frac{-10+8}{-1}=2$

$\therefore$ The harmonic conjugate of P is $(-3,-4,2)$