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Chapter 12 Conics

Chapter 12 Conics. Section 2 Circles. Section 12.2 Objectives. 1 Write the Standard Form of the Equation of a Circle 2 Graph a Circle 3 Find the Center and Radius of a Circle from an Equation in General Form. Axis. Axis. Axis. Axis. Parabola. Ellipse. Circle. Hyperbola.

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Chapter 12 Conics

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  1. Chapter 12 Conics Section 2 Circles

  2. Section 12.2 Objectives 1 Write the Standard Form of the Equation of a Circle 2 Graph a Circle 3 Find the Center and Radius of a Circle from an Equation in General Form

  3. Axis Axis Axis Axis Parabola Ellipse Circle Hyperbola Conic Sections Conics, an abbreviation for conic sectionsare curves that result from the intersection of a right circular cone and a plane. The four conics are shown below.

  4. r (h, k) (x, y) Radius and Center • Acircleis a set of all points in the Cartesian plain that are a fixed distance r from a fixed point (h, k). The fixed distance r is called the radius, and the fixed point (h, k) is called the center of the circle.

  5. Standard Form of a Circle The standard form of an equation of a circle with radius r and center (h, k) is (x – h)2 + (y – k)2 = r2. Example: Determine the equation of the circle with radius 4 and center (– 5, 2). (x – h)2 + (y – k)2 = r2 center (– 5, 2) r = 4 (x – (– 5))2 + (y – (2))2 = 42 (x + 5)2 + (y – 2)2 = 16

  6. y 8 (– 5, 6) 6 4 (– 5, 2) 2 (– 1, 2) (– 9, 2) x 8 8 6 4 2 2 4 6 (– 5, – 2) 4 6 8 Graphing a Circle Example: Graph the equation (x + 5)2 + (y – 2)2 = 16. h = – 5, k = 2 r = 4 The center is (– 5, 2). The radius is 4.

  7. General Form of a Circle The general form of the equation of a circle is given by the equation x2 + y2 + ax + by + c = 0 when the graph exists. Example: Determine the equation of the circle: x2 + y2 + 2x – 8y + 8 = 0 Regroup the terms. (x2 + 2x) + (y2 – 8y) = – 8 (x2 + 2x + 1) + (y2 – 8y + 16) = – 8 + 1 + 16 Complete the square in both x and y. (x+ 1)2 + (y – 4)2 = 9 Factor.

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