The area inside the conic described by the equation x^2 + 6x = 4y - y^2 is 13π, as it represents a circle in standard form with a center at (-3, 2) and a radius of √13.
To find the area inside the conic described by the equation x^2 + 6x = 4y - y^2, we can start by completing the square to rewrite the equation in a standard form. The standard form of a conic section equation is usually written as Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0, where A, B, C, D, E, and F are constants.
Let's complete the square for the given equation:
x^2 + 6x = 4y - y^2
Move all terms to one side of the equation:
x^2 + 6x + y^2 - 4y = 0
Now, complete the square for both the x-terms and y-terms. For the x-terms:
x^2 + 6x + 9
For the y-terms:
y^2 - 4y + 4
Add these completing the square terms to both sides of the equation:
x^2 + 6x + 9 + y^2 - 4y + 4 = 9 + 4
Now, factor the perfect square trinomials:
(x + 3)^2 + (y - 2)^2 = 13
This equation represents a circle in standard form, where the center of the circle is (-3, 2), and the radius is √(13).
Now, to find the area inside the circle, we use the formula for the area of a circle:
A = π r^2
where r is the radius. In this case, the radius is √(13). Therefore, the area is:
A = π (√(13))^2 = 13π
So, the area inside the given conic section is 13π.