ng>First Circle:

$ 2x^2 + 2y^2 - 2x + 6y - 3 = 0 $

Second Circle:

$ x^2 + y^2 + 4x + 2y + 1 = 0 $

Subtract the second circle from the first:

$ (x^2 + y^2 + 4x + 2y + 1) - (2x^2 + 2y^2 - 2x + 6y - 3) = 0 $

Simplifying, we have:

$ x^2 + y^2 + 4x + 2y + 1 - x^2 - y^2 + x - 3y + \frac{3}{2} = 0 $

This simplifies to:

$ 5x - y + \frac{5}{2} = 0 $

To eliminate the fraction, multiply the entire equation by 2:

$ 10x - 2y + 5 = 0 \tag{i} $

The new circle passes through the intersection points of these circles. Its equation is of the form:

$ (x^2 + y^2 - x + 3y - \frac{3}{2}) + \mu(x^2 + y^2 + 4x + 2y + 1) = 0 \tag{ii} $

Let’s simplify this:

$ x^2 + y^2 + \frac{(4\mu - 1)}{(1 + \mu)}x + \frac{(3 + 2\mu)}{(1 + \mu)}y + \mu - \frac{3}{2} = 0 $

The center $(h, k)$ of the circle is given by:

$ \left( \frac{1 - 4\mu}{2(1+\mu)}, \frac{-(3 + 2\mu)}{2(1+\mu)} \right) $

Substitute into the chord equation (Equation (i)):

$ 5 \left( \frac{1 - 4\mu}{1+\mu} \right) + \left( \frac{3 + 2\mu}{1+\mu} \right) + 5 = 0 $

Simplifying, we get:

$ 5 - 20\mu + 3 + 2\mu + 5 + 5\mu = 0 $

This results in:

$ 13 - 13\mu = 0 $

Thus:

$ \mu = 1 $

Substitute $\mu = 1$ back into Equation (ii):

$ 2x^2 + 2y^2 + 3x + 5y - \frac{1}{2} = 0 $

Multiply the entire equation by 2 to clear the fraction:

$ 4x^2 + 4y^2 + 6x + 10y - 1 = 0 $

This is the equation of the required circle.

Circle

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