Question

The equation of the circle concentric with x² + y² − 3x + 4y − c = 0 and passing through (−1, −2) is

Options

•  x² + y² − 3x + 4y − 1 = 0

• x² + y² − 3x + 4y = 0

• x² + y² − 3x + 4y + 2 = 0

• none of these

Answer 1

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):

\[\left( x - \frac{3}{2} \right)^2 + \left( y + 2 \right)^2 = \left( \frac{5}{2} \right)^2 \]

\[ \Rightarrow \frac{\left( 2x - 3 \right)^2}{4} + \left( y + 2 \right)^2 = \frac{25}{4}\]

\[ \Rightarrow \left( 2x - 3 \right)^2 + 4 \left( y + 2 \right)^2 = 25\]

\[ \Rightarrow x^2 + y^2 - 3x + 4y = 0\]

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