Question

The locus of feet of perpendiculars drawn from the origin to the straight lines passing through $(2,1)$ is

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

Answer 1

The correct option is B $x^2+y^2-2x-y=0$

Let coordinates of the feet be $(x,y).$
Slope of the perpendiculars from the origin $=\frac{y-0}{x-0}$
Slope of the line passing through $(2,1)$ and feet of the perpendiculars from the origin $=\frac{y-1}{x-2}$
Since the lines are perpendicular,
$\left(\frac{y-0}{x-0}\right)\left(\frac{y-1}{x-2}\right)=-1$
$\Rightarrow x^2+y^2-2x-y=0$