Question

A point $P$ is such that its perpendicular distance from the line $y-2x+1=0$ is equal to its distance from the origin,then the locus of the point $P.$

• A. $x^2+4y^2+4xy+4x-2y-1=0$
• B. $x^2+y^2+4xy+x-y-1=0$
• C. $x^2+4y^2-1=0$
• D. None of these

Answer 1

The correct option is A $x^2+4y^2+4xy+4x-2y-1=0$

Let the coordinates of point $P$ be $(h,k)$
$\frac{|k-2h+1|}{\sqrt{5}}=\sqrt{h^2+k^2}$
$\Rightarrow (k-2h+1{)}^2=5(h^2+k^2)$
$\Rightarrow h^2+4k^2+4hk+4h-2k-1=0$
$\Rightarrow x^2+4y^2+4xy+4x-2y-1=0$