Thank you for reporting, we will resolve it shortly
Q.
A chord $AB$ is drawn from the point $A (0, 3)$ on the circle $x^2 + 4x + (y - 3)2 = 0$, and is extended to $M$ such that
$AM = 2AB$. The locus of $M$ is
Given, $AM = 2AB$
$\Rightarrow B$ is mid-point of $A M$.
$\therefore $ Coordinate of $B$ is $\left(\frac{0+x_{1}}{2}, \frac{3+y_{1}}{2}\right)$
$=\left(\frac{x_{1}}{2}, \frac{y_{1}+3}{2}\right)$
Since, $B$ lies on the circle $x^{2}+4 x+(y-3)^{2}=0$
$\therefore \left(\frac{x_{1}}{2}\right)^{2}+4\left(\frac{x_{1}}{2}\right)+\left(\frac{y_{1}+3}{2}-3\right)^{2}=0$
$\Rightarrow \frac{x_{1}^{2}}{4}+2 x_{1}+\left(\frac{y_{1}-3}{2}\right)^{2}=0$
$\Rightarrow \frac{x_{1}^{2}}{4}+2 x_{1}+\frac{y_{1}^{2}+9-6 y_{1}}{4}=0$
$\Rightarrow x_{1}^{2}+y_{1}^{2}+8 x_{1}-6 y_{1}+9=0$
Hence, locus of a point is
$x^{2}+y^{2}+8 x-6 y+9=0$
