Chapter 12Conics12.1 Distance & Midpoint Formulas12.2 Circles

Sullivan, III & Struve, Elementary and Intermediate Algebra12.1 - *Copyright 2010 Pearson Education, Inc.

Section 12.1 Objectives

1Use the Distance Formula2Use the Midpoint Formula

Sullivan, III & Struve, Elementary and Intermediate Algebra12.1 - *Copyright 2010 Pearson Education, Inc.

The Distance Formula

The Distance FormulaThe distance between two points P1 = (x1, y1) and P2 = (x2, y2), is

Sullivan, III & Struve, Elementary and Intermediate Algebra12.1 - *Copyright 2010 Pearson Education, Inc.

The Distance Formula

Example:Determine the distance between (3, 5) and (3, 3).

Sullivan, III & Struve, Elementary and Intermediate Algebra12.1 - *Copyright 2010 Pearson Education, Inc.

Example:Find the distance between ( 6, 6) and ( 5, 2).The Distance Formula

Sullivan, III & Struve, Elementary and Intermediate Algebra12.1 - *Copyright 2010 Pearson Education, Inc.

The midpoint of a line segment is the point located exactly halfway between the two endpoints of the line segment.The Midpoint Formula

Sullivan, III & Struve, Elementary and Intermediate Algebra12.1 - *Copyright 2010 Pearson Education, Inc.

Example:Find the midpoint of the line segment joining P1 = (0, 8) and P2 = (4, 6). M = (2, 1)The Midpoint Formula

Chapter 12ConicsSection 2Circles

Sullivan, III & Struve, Elementary and Intermediate Algebra12.1 - *Copyright 2010 Pearson Education, Inc.

Section 12.2 Objectives

1Write the Standard Form of the Equation of a Circle 2Graph a Circle3Find the Center and Radius of a Circle from an Equation in General Form

Sullivan, III & Struve, Elementary and Intermediate Algebra12.1 - *Copyright 2010 Pearson Education, Inc.

Conic SectionsConics, an abbreviation for conic sections are curves that result from the intersection of a right circular cone and a plane. The four conics are shown below.

Sullivan, III & Struve, Elementary and Intermediate Algebra12.1 - *Copyright 2010 Pearson Education, Inc.

Radius and CenterA circle is 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.

Sullivan, III & Struve, Elementary and Intermediate Algebra12.1 - *Copyright 2010 Pearson Education, Inc.

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.(x h)2 + (y k)2 = r2(x ( 5))2 + (y (2))2 = 42r = 4center ( 5, 2)(x + 5)2 + (y 2)2 = 16Example:Determine the equation of the circle with radius 4 and center ( 5, 2).

Sullivan, III & Struve, Elementary and Intermediate Algebra12.1 - *Copyright 2010 Pearson Education, Inc.

Graphing a Circle

Example:Graph the equation (x + 5)2 + (y 2)2 = 16.The center is ( 5, 2).h = 5, k = 2 r = 4 The radius is 4.

Sullivan, III & Struve, Elementary and Intermediate Algebra12.1 - *Copyright 2010 Pearson Education, Inc.

General Form of a Circle

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