Question

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

• A. $x^2+y^2-8x-6y+9=0$
• B. $x^2+y^2+8x+6y+9=0$
• C. $x^2+y^2+8x–6y+9=0$
• D. $x^2+y^2–8x+6y+9=0$

Answer 1

The correct option is C $x^2+y^2+8x–6y+9=0$

Let the coordinates of $M(h,k)$
Then the coordinates of
$B=(\frac{h+0}{2},\frac{k+3}{2})$
As $B$ lies on the circle, so
$\frac{h^2}{4}+2h+\frac{(k-3{)}^2}{4}=0\Rightarrow h^2+8h+k^2-6k+9=0$

$\therefore$ The locus of M is $x^2+y^2+8x–6y+9=0$