Question

A pair of perpendicular straight lines is drawn through the origin forming with the line $2x+3y=6$ an isosceles triangle right angled at the origin. The equation to the line pair is

• A. $5x^2-24xy-5y^2=0$
• B. $5x^2-26xy-5y^2=0$
• C. $5x^2+24xy-5y^2=0$
• D. $5x^2+26xy-5y^2=0$

Answer 1

Answer: (A)

$5x^2-24xy-5y^2=0$

Let the lines be $y-mx=0$ and $my+x=0$

Therefore for $y=\frac{-2x}{3}+2$

We get
$\tan {45}^{\circ }=\left|\frac{m+\frac{2}{3}}{1-\frac{2m}{3}}\right|$

$1=\pm \left(\frac{3m+2}{3-2m}\right)$

Therefore
$3-2m=3m+2$

$5m=1$

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

and $3-2m=-3m-2$

$m=-5$

Therefore the equation of the lines are
$(y+5x)(-5y+x)=0$

$-5y^2+xy-25xy+5x^2=0$

$5x^2-24xy-5y^2=0$