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 $AB$ is drawn and extended to a point $M$, such that $AM=2AB$. An equation of the locus of $M$ is

• A. $x^2+6x+(y-2{)}^2=0$
• B. $x^2+8x+(y-3{)}^2=0$
• C. $x^2+y^2+8x-6y+9=0$
• D. $x^2+y^2+6x-4y+4=0$

Answer 1

Answer: (B), (C)

The correct options are
B $x^2+8x+(y-3{)}^2=0$
C $x^2+y^2+8x-6y+9=0$

Given equation of circle is $x^2+4x+{(y-3)}^2=0,AM=2AB$

$B$ is the mid-point of $AM$

$\Rightarrow B=(\frac{h}{2},\frac{k+3}{2})$ lies on the circle

equation of circle is $x^2+4x+{(y-3)}^2=0$

Let $x=\frac{h}{2},y=\frac{k+3}{2}$

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

$\frac{h^2}{4}+2h+\frac{k^2-6k+9}{4}=0$

$\therefore k^2+h^2+8h-6k+9=0$

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