Suppose that the points $\left(h,k\right),\left(1,2\right)$ and $\left(-3,4\right)$ lie on the line $L_1.$ If a line $L_2$ passing through the points $\left(h,k\right)$ and $\left(4,3\right)$ is perpendicular to $L_1,$ then $\frac{k}{h}$ equals:

• A

  $-\frac{1}{7}$
• B

  $3$
• C

  $0$
• Correct answer
  D

  $\frac{1}{3}$

Solution

Slope of line $L_1$ is $\frac{4-2}{-3-1}=\frac{2}{-4}=\frac{-1}{2}$
Hence slope of $L_2$ is $2.$
From line $L_1, \frac{k-2}{h-1}=\frac{-1}{2}$
$\Rightarrow h+2k=5$ …..(1)
From line $L_2, \frac{k-3}{h-4}=2$
$\Rightarrow k=2h-5$ ..…(2)
From (1) and (2) $h=3$ and $k=1$
$\Rightarrow \frac{k}{h}=\frac{1}{3}$

Hence, the correct answer is option: 'D'.