Question

Which of the following equations represents a circle with centre $(g,-2f)$ and radius $\sqrt{2g^2+4f^2-c}$ ?

• A. $x^2+y^2+2gx+2fy+c-f^2=0$
• B. $x^2+y^2-2gx-2fy+c+g^2=0$
• C. $x^2+y^2-2gx+4fy-g^2+c=0$
• D. $x^2+y^2-2gx+4fy-g^2-c=0$

Answer 1

The correct option is C $x^2+y^2-2gx+4fy-g^2+c=0$

Equation of a circle with centre $(h,k)$ and radius r is given by
$(x-h{)}^2+(y-k{)}^2=r^2$

Given, the centre is $(g,-2f)$ and radius is $\sqrt{2g^2+4f^2-c}$

$\Rightarrow (x-g{)}^2+(y+2f{)}^2=(\sqrt{2g^2+4f^2-c}{)}^2$

$\Rightarrow x^2+y^2-2gx+4fy+g^2+4f^2=2g^2+4f^2-c$

$\Rightarrow x^2+y^2-2gx+4fy-g^2+c=0$