Question

The line through the points $(h,3)$ and $(4,1)$ intersects the line $7x-9y-19=0$ at right angle. Find the value of $h$.

Answer 1

The slope of line passing through points $(h,3)$ and $(4,1)$ is,
$m_1=\frac{1-3}{4-h}=\frac{-2}{4-h}$
and the slope of line $7x-9y-19=0$ is $m_2=\frac{7}{9}$
It is given that the two lines are perpendicular
$\therefore m_1\cdot m_2=-1$
$\Rightarrow \left\{\frac{-2}{4-h}\right\}\cdot \left\{\frac{7}{9}\right\}=-1$
$\Rightarrow \frac{-14}{36-9h}=-1$
$\Rightarrow 9h=36-14$
$\Rightarrow h=\frac{22}{9}$