If $(2, - 1)$ lies on the circle $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ which is concentric with the circle $x^{2} + y^{2} + 4 x - 6 y + 3 = 0$ , then $c$ is

19

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Solution

The correct option is B $- 19$
The general equation of circle is $x^{2} + y^{2} + 2 m x + 2 n y + z = 0$
where,
Center = $(- m, - n)$
Radius = $\sqrt{m^{2} + n^{2} - z}$

So, For $x^{2} + y^{2} + 4 x - 6 y + 3 = 0$ ,
Center = $(- 2, 3)$

Since the circles are concentric,
$x^{2} + y^{2} + 2 g x + 2 f y + c = 0$
$\to x^{2} + y^{2} + 4 x - 6 y + c = 0$

Now, $(2, - 1)$ lies on $x^{2} + y^{2} + 4 x - 6 y + c = 0$
$\to 4 + 1 + 8 + 6 + c = 0$
$\to c = - 19$

Hence, (B) is the correct answer.

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If (2,−1) lies on the circle x2+y2+2gx+2fy+c=0 which is concentric with the circle x2+y2+4x−6y+3=0, then c is

If $(2, - 1)$ lies on the circle $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ which is concentric with the circle $x^{2} + y^{2} + 4 x - 6 y + 3 = 0$ , then $c$ is