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

Answer: 4

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