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.

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Presentation transcript:

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.1 Derive 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 Geometry Section 1: Circles in the coordinate Plane

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 = r 2.

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 Square Recall that we can solve quadratic equations of the form ax 2 + bx + c = 0 by “completing the square”… 1. Subtract c from both sides. ax 2 + bx = - c 2. Square half the coefficient of x and add it to both sides. ax 2 + bx + = - c + 3. Then “un-FOIL” We can use this process to find the standard equation of circles when given the “FOILed” equation.

Completing the Square to Find the Equation of a Circle Find the center and radius of the circle whose equation is x 2 + y 2 – 8x + 6y + 5 = 0.

 Find the center and radius of a circle whose equation is x 2 + y 2 + 8x – 6y – 15 = 0.