Question

A point $P$ lies on the line whose end points are $A(1,2,3)$ and $B(2,10,1).$
If $z$-co-ordinate of $P$ is $7$, find sum of its other co - ordinates.

Answer 1

Let $P$ divide $AB$ in the ratio $\lambda :1$
$\therefore P(\frac{2\lambda +1}{\lambda +1},\frac{10\lambda +2}{\lambda +1},\frac{\lambda +3}{\lambda +1})$
But $z$-co-ordinate of $P$ is $7$
$\therefore \frac{\lambda +3}{\lambda +1}=7$ or $\lambda =-\frac{2}{3}$
Putting for $\lambda$ we get the point $P$ as $(-1,-14,7)$

Therefore, sum of the co-ordinates is $-8$.