Question

From the point $A\left(0.3\right)$ 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

• A. $x^2+8x+y^2=0$
• B. $x^2+8x+(y-3{)}^2=0$
• C. $(x-3{)}^2+8x+y^2=0$
• D. $x^2+8x+8y^2=0$

Answer 1

The correct option is B $x^2+8x+(y-3{)}^2=0$

Given equation of circle is

$x^2+4x+(y-3{)}^2=0$

Now $A(0,3)$ is on the circle

Let point $M$ be $(h,k)$

As $AM=2AB$

$AB=\frac{AM}{2}$

Since $B$ is the midpoint of $AM$

Coordinates of $B$ are $(\frac{h+0}{2},\frac{k+3}{2})=(\frac{h}{2},\frac{k+3}{2})$

$B$ lies on circle

$\Rightarrow \frac{h^2}{4}+4,\frac{h}{2}+{(\frac{k+3}{2}-3)}^2=0$

$\Rightarrow \frac{h^2}{4}+2h+{\left(\frac{k-3}{2}\right)}^2=0$

$\frac{h^2}{4}+8h+(k-3{)}^2=0$

Hence equation of locus is

$x^2+8x+(y-3{)}^2=0$