Mathematics - Circle Question with Solution | TestHub

For centre $(x,y)$ and radius $|x|$, external tangency gives $(x+1{)}^2+y^2=(|x|+1{)}^2$, i.e. $2x+y^2=2|x|$. For $x<0$ this is $y^2+4x=0$; for $x\ge 0$ it forces $y=0$. Hence the locus is $\{y^2+4x=0\}\cup \{y=0,x\ge 0\}$.