Question

From a point $A(0,3)$ on the circle $x^2+4x+(y-3{)}^2=0$, a chord $AB$ is drawn and extended to a point $M$ such that $AM=2AB$. Find the equation of the locus of $M$.

Answer 1

Since $AM=2AB$, we have $AB=\frac{1}{2}AM$.
It follows that $B$ is the mid-point of $AM$. If $M$ is the point $(h,k)$, then $B$ is $\left[\frac{1}{2}\right(h+0),\frac{1}{2}(k+3\left)\right]$ i.e. $[\frac{1}{2}h,\frac{1}{2}(k+3\left)\right]$. But $AB$ is a chord of the circle $x^2+4x+(y-3{)}^2=0$, and hence the point $B$ lies on the circle.
$\therefore {\left(\frac{1}{2}h\right)}^2+4\left(\frac{1}{2}h\right)+{\left[\frac{1}{2}\right(k+3)-3]}^2=0$
or $h^2+k^2+8h-6k+9=0$
$\therefore$ The locus of $M$ is the circle
$x^2+y^2+8x-6y+9=0$.