Question

A chord $AB$ drawn from the point $A(0,3)$ at circle $x^2+4x+(y-3{)}^2=0$ and it meets to $M$ in such a way that $AM=2AB$, then the locus of point $M$ will be

• A. Straight line
• B. Circle
• C. Parabola
• D. Hyperbola

Answer 1

Answer: (B)

Circle

$AM=2AB$ means that $B$ is the midpoint of $AM$.

Let $M=(h,k)$, then $B=(\frac{h}{2},\frac{k+3}{2})$.

Since $AB$ is a chord, $B$ also lies on the circle.

On substituting the coordinates of $B$ in the equation of the circle, we get

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

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

Clearly therefore, the locus of $M$ is a circle.

So option B is the correct answer.