Warm UP

• Solve using the Pythagorean theorem

ESSENTIAL QUESTION:

• How can you write an equation for a circle in the coordinate plane with known center and radius?

MCC9-12.G.GPE.1Derive the equation of a circle of given center

and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation

Unit 6: Coordinate Unit 6: Coordinate GeometryGeometry

Section 1: Circles in the coordinate Plane

Vocabulary Vocabulary

• A circle is the set of all points in a plane that are equidistant from a reference point in that plane, called the center. The set of points forms a 2-dimensional curve that measures 360°.

• The center of a circle is the point in the plane of the circle from which all points on the circle are equidistant. The center is in the interior of the circle.

• The radius of a circle is the distance from the center to a point on the circle.

Equation of a Circle

Standard Equation of a Circle:An equation of a circle with center (h, k) and radius r is (x – h)2 + (y – k)2 = r2.

Writing the Equation of a Circle

• What is the standard equation of a circle with center (5, -2) and radius 7?

What is the standard equation of each circle? • Center (3, 5); radius 6

• Center (-2, -1); radius 2

Using the Center and a Point on a Circle

• What is the standard equation of the circle with center (1, -3) that passes through the point (2, 2)?

What is the standard equation of the circle with center (5, 3) that passes through the point (-1, 1)?

Graphing a Circle Given Its Equation

• Graph the circle whose equation is (x + 4)2 + (y – 1)2 = 25.

Completing the SquareRecall that we can solve quadratic equations of the form ax2 + bx + c = 0 by “completing the square”…

1. Subtract c from both sides. ax2 + bx = - c

2. Square half the coefficient of x and add it to both sides.

ax2 + bx + = - c +

3. Then “un-FOIL”

We can use this process to find the standard equation of circles when given the “FOILed” equation.

2

2

b

2

2

b

Completing the Square to Find the Equation of a Circle

Find the center and radius of the circle whose equation is x2 + y2 – 8x + 6y + 5 = 0.

Find the center and radius of a circle whose equation is x2 + y2 + 8x – 6y – 15 = 0.