Circles - Complete the Square

Front

What is the standard form of a circle's equation?

Back

The standard form of a circle's equation is (x - h)² + (y - k)² = r², where (h, k) is the center of the circle and r is the radius.

Tags

CCSS.HSG.GPE.A.1

Front

How do you complete the square for the equation x² + 6x + y² - 12y + 20 = 0?

Back

To complete the square, rearrange the equation: (x² + 6x) + (y² - 12y) = -20. Then, complete the square for x and y: (x + 3)² - 9 + (y - 6)² - 36 = -20. This simplifies to (x + 3)² + (y - 6)² = 25.

Tags

CCSS.HSG.GPE.A.1

Front

What is the center of the circle given by the equation (x + 3)² + (y - 6)² = 25?

Back

The center of the circle is (-3, 6).

Tags

CCSS.HSG.GPE.A.1

Front

What is the radius of the circle given by the equation (x + 3)² + (y - 6)² = 25?

Back

The radius of the circle is 5, since r² = 25, thus r = √25 = 5.

Tags

CCSS.HSG.GPE.A.1

Front

How do you find the radius from the equation (x + 2)² + (y - 3)² = 36?

Back

The radius is found by taking the square root of the right side of the equation: r = √36 = 6.

Tags

CCSS.HSG.GPE.A.1

Front

What does the equation x² + y² + 6x - 2y + 1 = 0 represent?

Back

This equation represents a circle, which can be rewritten in standard form by completing the square.

Tags

CCSS.HSG.GPE.A.1

Front

What is the process to determine the center and radius from the equation x² + y² - 10x + 8y - 8 = 0?

Back

Rearrange the equation to group x and y terms, complete the square for both, and then rewrite in standard form to find the center (5, -4) and radius 7.

Tags

CCSS.HSG.GPE.A.1

Front

What is the significance of the terms (h, k) in the standard form of a circle's equation?

Back

The terms (h, k) represent the coordinates of the center of the circle.

Tags

CCSS.HSG.GPE.A.1

Front

How can you identify the radius from the standard form of a circle's equation?

Back

The radius can be identified as the square root of the constant term on the right side of the equation: (x - h)² + (y - k)² = r².

Tags

CCSS.HSG.GPE.A.1

Front

What is the first step in converting a general circle equation to standard form?

Back

The first step is to rearrange the equation to group the x terms and y terms together.

Tags

CCSS.HSG.GPE.A.1