Question

If $(4,-2)$ is a point on the circle $x^2+y^2+2gx+2fy+c=0$ which is concentric to $x^2+y^2-2x+4y+20=0$ then value of $c$ is

• A. $-4$
• B. $0$
• C. $4$
• D. $1$

Answer 1

Answer: (A)

$-4$

Since the circle $S_1\equiv 3x^2+y^2+2gx+2fy+c=0$ is concentric to $S_2\equiv x^2+y^2-2x+4y+20=0$

$\therefore$ both have same centre.

centre of $S_2=(\frac{-1}{2}\times -2,\frac{1}{2}\times 4)=(-1,2)$

$\therefore$ for $S_1$ $g=-1$ $f=2$

$\therefore$ Equation of circle $S_1$ is $x^2+y^2-2x+4y+c=0$ as it passes through $(4,-2)$

$\therefore x^2+y^2-2x+4y+c=0\Rightarrow 4^2+{(-2)}^2-2\times 4+4\times -2+c=0\Rightarrow 16+4-8-8+c=0\Rightarrow c=-4$