## Right Angle

If the graphs of the equations $3x-5y+4=0$ and $2x+ay-11=0$ intersect at right angles, find the value of $a$.
Source: NCTM Math Teacher 2008

SOLUTION
$3x-5y+4=0$
$5y=3x+4$
$y=\dfrac{3}{5}x+\dfrac{4}{5}$

$2x+ay-11=0$
$ay=-2x+11$
$y=-\dfrac{2}{a}x+\dfrac{11}{a}$
If the two graphs intersect at right angle, their slopes are negative reciprocal of each other
$-\dfrac{2}{a}=-1\div\dfrac{3}{5}$
$\dfrac{2}{a}=\dfrac{5}{3}$
Cross multiply
$5a=6$
$a=\dfrac{6}{5}$
Their graphs are shown below

Answer: $\dfrac{6}{5}$