Suppose that the points $(\mathrm{h}, \mathrm{k}),(1,2)$ and $(-3,4)$ lie on the line $l_1$. If a line $l_2$…
Suppose that the points $(\mathrm{h}, \mathrm{k}),(1,2)$ 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 $\left(\frac{k}{h}\right)$ equals
$\frac{1}{3}$
0
3
$-\frac{1}{7}$
Solution
Slope of line $l_1=\frac{4-2}{-3-1}=-\frac{1}{2}$
Equation of $l_1$ is
$y-4=-\frac{1}{2}(x+3)$
$\Rightarrow x+2 y=5$...(i)
Since $l_1 \perp l_2$
$\therefore \quad$ Slope of $l_2=2$
Equation of $l_2$ is
$\begin{aligned}
& y-3=2(x-4) \\
& \Rightarrow 2 x-y=5...(ii)
\end{aligned}$
Solving (i) and (ii), we get
$\begin{aligned}
& x=3, y=1 \\
& \Rightarrow \mathrm{~h}=3, \mathrm{k}=1 \\
& \Rightarrow \frac{\mathrm{k}}{\mathrm{~h}}=\frac{1}{3}
\end{aligned}$