Question

The centre and radius of the circle $x^2+y^2+2gx+2yf+c=0$ are

• A. $(g,f)$ and $\sqrt{g^2+f^2-c}$
• B. $(-g,-f)$ and $\sqrt{g^2+f^2-c}$
• C. $(g,f)$ and $\sqrt{g^2+f^2+c}$
• D. $(-g,-f)$ and $\sqrt{g^2+f^2+c}$

Answer 1

The correct option is B

$(-g,-f)$ and $\sqrt{g^2+f^2-c}$

The general equation of a circle is given $x^2+y^2+2gx+2yf+c=0$ . The centre and radius of this circle are $(-g,-f)$ and $\sqrt{g^2+f^2-c}$ respectively ; We will try to get this result.

We know the equation of a circle whose centre is $(h,k)$ and radius r, is $(x-h{)}^2+(y-k{)}^2=r^2$

We will simplify this and compare it with $x^2+y^2+2gx+2yf+c=0$ to find radius and centre.

$\Rightarrow x^2+y^2-2hx-2ky+h^2+k^2-r^2=0$ - - - - - (1)

$x^2+y^2+2gx+2yf+c=0$ - - - - - (2)

(1) and (2) represents the same circle.

$\Rightarrow$ -h = g and -k = f

$\Rightarrow (h,k)\equiv (-g,-f)$is the centre.

We also have $c=h^2+k^2-r^2=0$

$(h,k)=(-g,-f)$

$\Rightarrow c=(-g{)}^2+(-f{)}^2-r^2$

$\Rightarrow r=\sqrt{g^2+f^2-c}$