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Q.
The line through the points $(h, 3)$ and $(4,1)$ intersects the line $7 x-9 y-19=0$ at right angle. Then, the value of $h$ is
Straight Lines
Solution:
Slope of the line passing through $(h, 3)$ and $(4,1)$ is
$m_1=\frac{1-3}{4-h}=\frac{-2}{4-h} \left(\because m=\frac{y_2-y_1}{x_2-x_1}\right)$
Given, equation of line is
$7 x-9 y-19 =0$
$y =\frac{7}{9} x-\frac{19}{9} ....$(i)
On comparing Eqs. (i) with $y=m x+c$, we get
$m_2=\frac{7}{9}$
$\because$ The lines are perpendicular to each other.
$ \therefore m_1 m_2 =-1 $
$\Rightarrow -\frac{2}{4-h} \times \frac{7}{9} =-1$
$\Rightarrow 14 =9(4-h) $
$\Rightarrow 14=36-9 h \Rightarrow 9 h=22 $
$\therefore h =\frac{22}{9}$