Question

If the distance of $P$ from the origin is twice the distance from $(1,2)$, then the equation of the locus of $P$ is

• A. $3(x^2+y^2)-8x-16y+20=0$
• B. $3(x^2+y^2)-8x+16y-20=0$
• C. $x^2+y^2+8x+16y+20=0$
• D. $x^2+y^2-8x-16y-20=0$

Answer 1

The correct option is A $3(x^2+y^2)-8x-16y+20=0$

Consider an arbitrary point $P(x,y)$.
Then, $OP=\sqrt{(x-0{)}^2+(y-0{)}^2}$
$=\sqrt{x^2+y^2}$...(i)
Distance of $P$ from the point $(1,2)$ is
$d^{\prime }=\sqrt{(x-1{)}^2+(y-2{)}^2}$ ...(ii)
Now it is given that
$OP=2d^{\prime }$
$O{P}^2=4d^{\prime 2}$
$\Rightarrow x^2+y^2=4\left(\right(x-1{)}^2+(y-2{)}^2)$
$\Rightarrow x^2+y^2=4(x^2+y^2-2x-4y+5)$
$\Rightarrow 3x^2+3y^2-8x-16y+20=0$