Question

Suppose that the points $(h, k), (1, 2)$ and $(-3 ,4)$ lie on the line $L_1$ If a line $L_2$ passing through the points $(h, k)$ and $(4, 3)$ is perpendicular on $L_1$, then $k + h$ equal to

Answer 1

$\because (h, k), (1, 2)$ and $(-3, 4)$ are collinear
$\therefore \begin{vmatrix}h&k&1\\ 1&2&1\\ -3&4&1\end{vmatrix} = 0$
$ \Rightarrow - 2h - 4k + 10 = 0 $
$ \Rightarrow h + 2k = 5 .....\left(i\right) $
Now, $m_{L_1} = \frac{4-2}{-3-1} = -\frac{1}{2} $
$ \Rightarrow m_{L_2} = 2\left[\because L_{1} \bot L_{2}\right]$
By the given points $(h, k)$ and $(4, 3)$,
$m_{L_2} = \frac{k-3}{h-4} $
$ \Rightarrow \frac{k-3}{h -4} = 2 $
$\Rightarrow k - 3 = 2h - 8 $
$ 2h - k = 5 .....\left(ii\right) $
From (i) and (ii)
$k + h = 1 + 3 = 4$