If a pair of lines drawn through the origin forms an isosceles right-angled triangle with the line $2 x+3…
If a pair of lines drawn through the origin forms an isosceles right-angled triangle with the line $2 x+3 y=6$, then those lines are
$x-5 y=0,5 x+y=0$
$3 x-y=0, x+3 y=0$
$5 x-y=0, x+5 y=0$
$x-3 y=0,3 x+y=0$
Solution
Given line, $2 x+3 y=6$
Slope of line $=-2 / 3$
Equation of lines passing through origin and form
an isosceles right angle triangle with the given line
$\therefore$ Each angle be $45^{\circ}$.
$\therefore$ Equation of line passing through origin and
making an angle $45^{\circ}$ with the line $2 x+3 y=6$ is
$\frac{y}{x}=\frac{-2 / 3 \pm \tan 45^{\circ}}{1 \mp\left(\frac{-2}{3}\right) \tan 45^{\circ}}$
$\Rightarrow \quad \frac{y}{x}=\frac{-2 / 3+1}{1+\frac{2}{3}}$ or $\frac{y}{x}=\frac{-2 / 3-1}{1-2 / 3}$
$\Rightarrow \quad \frac{y}{x}=\frac{1}{5}$ or $\frac{y}{x}=\frac{-5}{1}$
$\Rightarrow \quad 5 y=x$ or $y+5 x=0$
$\Rightarrow \quad x-5 y=0$ or $5 x+y=0$