Question

complete the square and write the given equation in standard form. then give the center and radius of the circle and graph the equation. \\( x^2 + y^2 - 2x - 6y - 15 = 0 \\) the equation in standard form is \\( \square \\). (simplify your answer.) the center of the circle is \\( \square \\). (type an ordered pair.) the radius of the circle is \\( r = \square \\). use the graphing tool to graph the circle. click to enlarge graph

Explanation:

Step1: Group x and y terms

Rearrange the equation: \(x^2 - 2x + y^2 - 6y = 15\)

Step2: Complete the square for x

For \(x^2 - 2x\), take half of -2 (-1), square it (1). Add 1: \(x^2 - 2x + 1 = (x - 1)^2\)

Step3: Complete the square for y

For \(y^2 - 6y\), take half of -6 (-3), square it (9). Add 9: \(y^2 - 6y + 9 = (y - 3)^2\)

Step4: Add to both sides

Add 1 and 9 to the right: \((x - 1)^2 + (y - 3)^2 = 15 + 1 + 9 = 25\)

Step5: Identify center and radius

Standard form of a circle: \((x - h)^2 + (y - k)^2 = r^2\), so center \((h, k) = (1, 3)\), radius \(r = \sqrt{25} = 5\)

Answer:

The equation in standard form is \((x - 1)^2 + (y - 3)^2 = 25\).
The center of the circle is \((1, 3)\).
The radius of the circle is \(r = 5\).