Complete the square and write the equation in standard form. Then give the center and radius of each circle and graph the equation. x² + y² – 10x – 6y – 30 = 0

Verified step by step guidance

1. Start with the given equation: \(x^{2} + y^{2} - 10x - 6y - 30 = 0\).
2. Group the \(x\) terms and \(y\) terms together and move the constant to the other side: \(\left(x^{2} - 10x\right) + \left(y^{2} - 6y\right) = 30\).
3. Complete the square for the \(x\) terms: take half of \(-10\), which is \(-5\), then square it to get \$25\(. Add \)25\( inside the \)x$ group.
4. Complete the square for the \(y\) terms: take half of \(-6\), which is \(-3\), then square it to get \$9\(. Add \)9\( inside the \)y$ group.
5. Since you added \$25\( and \)9\( to the left side, add the same amounts to the right side to keep the equation balanced: \)30 + 25 + 9$.

Key Concepts

Completing the Square

Completing the square is a method used to rewrite quadratic expressions in the form (x - h)² or (y - k)² by adding and subtracting appropriate constants. This technique helps transform the general form of a circle's equation into its standard form, making it easier to identify key features like the center and radius.

Standard Form of a Circle's Equation

The standard form of a circle's equation is (x - h)² + (y - k)² = r², where (h, k) represents the center and r is the radius. Writing the equation in this form allows for straightforward identification of the circle's geometric properties and simplifies graphing.

Graphing Circles

Graphing a circle involves plotting its center (h, k) on the coordinate plane and using the radius r to mark points at a distance r in all directions. Understanding how to interpret the standard form equation aids in accurately sketching the circle and visualizing its position and size.