Question

If (−3, 2) lies on the circle x² + y² + 2gx + 2fy + c = 0 which is concentric with the circle x² + y² + 6x + 8y − 5 = 0, then c =
(a) 11
(b) −11
(c) 24
(d) none of these

Answer 1

(b) −11

The centre of the circle x² + y² + 6x + 8y − 5 = 0 is (−3, −4).

The circle x² + y² + 2gx + 2fy + c = 0 is concentric with the circle x² + y² + 6x + 8y − 5 = 0.
Thus, the centre of x² + y² + 2gx + 2fy + c = 0 is (−3, −4).
$\therefore g=3,f=4$

Also, it is given that (−3, 2) lies on the circle x² + y² + 2gx + 2fy + c = 0.

∴ ${\left(-3\right)}^2+2^2+2\left(3\right)\left(-3\right)+2\left(4\right)\left(2\right)+c=0$

$$\begin{gathered} \Rightarrow 9+4-18+16+c=0 \\ \Rightarrow c=-11 \end{gathered}$$