Questions: Identifying the center and radius to graph a circle given its equation in.. The equation of a circle is given below. Identify the radius and the center. Then graph the circle. 2 x^2 + 2 y^2 - 4 x + 12 y - 30 = 0 Radius: 5 Center: (1,-3)

Solution Steps

The given equation is: \[ 2x^2 + 2y^2 - 4x + 12y - 30 = 0 \]

First, divide the entire equation by 2 to simplify: \[ x^2 + y^2 - 2x + 6y - 15 = 0 \]

Group the x and y terms: \[ (x^2 - 2x) + (y^2 + 6y) = 15 \]

Complete the square for \(x\): \[ x^2 - 2x \rightarrow (x - 1)^2 - 1 \]

Complete the square for \(y\): \[ y^2 + 6y \rightarrow (y + 3)^2 - 9 \]

Substitute the completed squares back into the equation: \[ (x - 1)^2 - 1 + (y + 3)^2 - 9 = 15 \]

Combine constants on the right side: \[ (x - 1)^2 + (y + 3)^2 - 10 = 15 \] \[ (x - 1)^2 + (y + 3)^2 = 25 \]

Final Answer