Browse Questions

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

$\begin{array}{1 1}(A)\;h=-\large\frac{22}{9} \\(B)\; h=\large\frac{22}{9} \\(C)\; h=\large\frac{9}{22} \\(D)\;h=-\large\frac{9}{22} \end{array}$

Toolbox:
• Slope of a line passing through the points $(x_1,y_1)$ and $(x_2,y_2)$ is
• $m = \large\frac{y_2-y_1}{x_2-x_1}$
• If two lines are perpendicular, then the product of their slopes is -1.
Slope of the line passing through $(h,3)$ and $(4,1)$ is
$m_1=\large\frac{1-3}{4-h}$$=\large\frac{-2}{4-h} Equation of the given line is 7x-9y-19=0 This can be written as y = \large\frac{7}{9}$$x-\large\frac{19}{9}$
Hence the slope is $\large\frac{7}{9}$$\therefore m_2 = \large\frac{7}{9}. Since the lines are perpendicular, their product of the slopes is -1. (i.e.,) m_1m_2=-1 \Rightarrow \large\frac{-2}{4-h}$$ \times \large\frac{7}{9}$$=-1 \large\frac{-14}{36-9h}$$=-1$
$\Rightarrow -14 = -36+9h$
$\therefore 9h=22$
$\therefore h = \large\frac{22}{9}$