Question

The equation of the circle concentric with x² + y² − 3x + 4y − c = 0 and passing through (−1, −2) is
(a) x² + y² − 3x + 4y − 1 = 0
(b) x² + y² − 3x + 4y = 0
(c) x² + y² − 3x + 4y + 2 = 0
(d) none of these

Answer 1

(b) x² + y² − 3x + 4y = 0

The centre of the circle x² + y² − 3x + 4y − c = 0 is $\left(\frac{3}{2},-2\right)$.

Therefore, the centre of the required circle is $\left(\frac{3}{2},-2\right)$.
The equation of the circle is ${\left(x-\frac{3}{2}\right)}^2+{\left(y+2\right)}^2=a^2$. ...(1)


Also, circle (1) passes through (−1, −2).

$\therefore {\left(-1-\frac{3}{2}\right)}^2+{\left(-2+2\right)}^2=a^2$
⇒ $a=\frac{5}{2}$

Substituting the value of a in equation (1):

$$\begin{gathered} {\left(x-\frac{3}{2}\right)}^2+{\left(y+2\right)}^2={\left(\frac{5}{2}\right)}^2 \\ \Rightarrow \frac{{\left(2x-3\right)}^2}{4}+{\left(y+2\right)}^2=\frac{25}{4} \\ \Rightarrow {\left(2x-3\right)}^2+4{\left(y+2\right)}^2=25 \\ \Rightarrow x^2+y^2-3x+4y=0 \end{gathered}$$

Hence, the required equation of the circle is $x^2+y^2-3x+4y=0$.