Question

From the 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 $AM=2AB$. The equation of the locus of $M$ is....

Answer 1

Given, $(x+2{)}^2+(y-3{)}^2=4$
Let the coordinate be $M(h,k)$, where $B$ is mid-point of $A$ and $M$.
$\Rightarrow B(\frac{h}{2},\frac{k+3}{3})$
But $AB$ is the chord of circle
$x^2+4x+(y-3{)}^2=0$
Thus, $B$ must satisfy above equation.
$\therefore \frac{h^2}{4}+\frac{4h}{2}+[\frac{1}{2}(k+3)-3{]}^2=0$
$\Rightarrow h^2+y^2+8x-6y+9=0$
$\therefore$ Locus of $M$ is the circle
$x^2+y^2+8x-6y+9=0$