Question

A line is drawn a perpendicular to $5x=y+7$. Find the equation of the line if the area of the triangle formed by this line and the coordinate axes is $5$ sq units.

Answer 1

Slope of $5x=y+7$ is $5$

$\Rightarrow$ Let $m$ be slope of new line

$\Rightarrow m\times 5=-1$ (perpendicular)

$\Rightarrow m=-\frac{1}{5}$

$\Rightarrow$ line is of form $y=\frac{-x}{5}+c$

For $A,y=0$

$\Rightarrow x=5c$

For $B,x=0$

$\Rightarrow y=c$

$\Rightarrow A(5C,0),B(0,C)$

$area\left(OAB\right)=\frac{1}{2}\times OA\times OB$

$5=|\frac{1}{2}\times 5C\times C|$

$\Rightarrow \left|\frac{C^2}{2}\right|=1$

$\Rightarrow C^2=2$

$\Rightarrow C=\pm \sqrt{2}$

$\Rightarrow$ lines equation $y=\frac{-x}{5}\pm \sqrt{2}$.