Give the center and radius of the circle represented by each equation. x²+y²+6x+8y+9=0

Verified step by step guidance

1. Start with the given equation of the circle: \(x^2 + y^2 + 6x + 8y + 9 = 0\).
2. Group the \(x\) terms and \(y\) terms together: \((x^2 + 6x) + (y^2 + 8y) = -9\) (move the constant term to the right side).
3. Complete the square for the \(x\) terms: take half of 6, which is 3, then square it to get 9. Add 9 inside the \(x\) group and also add 9 to the right side to keep the equation balanced.
4. Complete the square for the \(y\) terms: take half of 8, which is 4, then square it to get 16. Add 16 inside the \(y\) group and also add 16 to the right side to keep the equation balanced.
5. Rewrite the equation as perfect square trinomials: \((x + 3)^2 + (y + 4)^2 = ext{(new constant on the right side)}\). From this form, identify the center as \((-3, -4)\) and the radius as the square root of the constant on the right side.

Key Concepts

Standard Form of a Circle's Equation

The standard form of a circle's equation is (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center and r is the radius. Converting a general quadratic equation into this form helps identify the circle's center and radius directly.

Completing the Square

Completing the square is a method used to rewrite quadratic expressions as perfect square trinomials. This technique is essential for transforming the given equation into the standard form of a circle by grouping x and y terms and completing the square for each.

Identifying the Center and Radius from the Equation

Once the equation is in standard form, the center (h, k) is found by taking the opposite sign of the values inside the parentheses, and the radius r is the square root of the constant on the right side. This allows for straightforward extraction of the circle's key features.