Mathematics - Circle Question with Solution | TestHub

The fixed circle has centre $(3,3)$ and radius 2. If the variable centre is $(x,y)$, its radius is $|x|$. External contact requires $(x-3{)}^2+(y-3{)}^2=(|x|+2{)}^2$. For $x\ge 0$ this gives $y^2-10x-6y+14=0$. The $x<0$ equation would force $x=\left((y-3{)}^2+5\right)/2>0$ and has no solutions, so D is the complete locus.