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