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 M such that AM =2AB. The equation of the locus of M is___

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

Answer 1

The correct option is D $x^2+y^2+8x-6y+9=0$

Given, $(x+2{)}^2+(y-3{)}^2=4$
Let the coordinate be M (h, k), where B is mid-point of A and M.
$\Rightarrow B(\frac{h}{2},\frac{k+3}{2})$
But AB is the chord of circle
$x^2+4x+(y-3{)}^2=0$
Thus, B must satisfy above equation.
$\therefore \frac{h^2}{4}+\frac{4h}{2}+{\left[\frac{1}{2}\right(k+3)-3]}^2=0$

$\Rightarrow h^2+k^2+8h-6k+9=0$
$\therefore$ Locus of M is the circle
$x^2+y^2+8x-6y+9=0$