Question

Find the equation of the circle concentric with the circle x² + y² − 6x + 12y + 15 = 0 and double of its area.

Answer 1

Let the equation of the required circle be

\[x^2 + y^2 + 2gx + 2fy + c = 0\] 

The centre of the circle x² + y² − 6x + 12y + 15 = 0 is (3, −6).

Area of the required circle = \[2\pi r^2\]

Here, r = radius of the given circle

Now, r = \[\sqrt{9 + 36 - 15} = \sqrt{30}\]

∴ Area of the required circle = \[2\pi\left( 30 \right) = 60\pi\]

Let R be the radius of the required circle.

∴\[60\pi = \pi R^2 \Rightarrow R^2 = 60\]

Thus, the equation of the required circle is

\[\left( x - 3 \right)^2 + \left( y + 6 \right)^2 = 60\]

\[x^2 + y^2 - 6x + 12y = 15\]