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 to $L_1$, then $\frac{k}{h}$ equals :

• A. $\frac{1}{3}$
• B. $3$
• C. $0$
• D. $-\frac{1}{7}$

Answer 1

The correct option is A $\frac{1}{3}$

Equation of line $L_1$ is
$y-2=\frac{4-2}{-3-1}(x-1)$
$\Rightarrow x+2y=5$
$(h,k)$ lies on line $L_1$.
$\Rightarrow h+2k=5\cdots \left(1\right)$

Equation of line perpendicular to $L_1$ is
$L_2:2x-y+c=0$
$(4,3)$ lies on line $L_2$.
$\Rightarrow 8-3+c=0\Rightarrow c=-5$
$L_2:2x-y=5$
$(h,k)$ lies on line $L_2$.
$\Rightarrow 2h-k=5\cdots \left(2\right)$

From $\left(1\right)$ and $\left(2\right)$,
$h=3,k=1$
$\therefore \frac{k}{h}=\frac{1}{3}$