5.2 Graph and Write 5.2 Graph and Write Equations of CirclesEquations of Circles

PgPg 180180

A circle is an infinite set of points in a plane that A circle is an infinite set of points in a plane that are equal distance away from a given fixed point are equal distance away from a given fixed point called a center.called a center.

A radius is a segment that connects the center A radius is a segment that connects the center and one of the points on the circle.and one of the points on the circle.

Every radius is equal in length. (radii)Every radius is equal in length. (radii)

A diameter is a line segment that connects two A diameter is a line segment that connects two points on the circle and goes through the center.points on the circle and goes through the center.

The formula for a circle with its center at (0,0) The formula for a circle with its center at (0,0) and a radius of “r” is:and a radius of “r” is:

Example: What is the equation for a circle whose Example: What is the equation for a circle whose center is at (0,0) and has a radius of 6. center is at (0,0) and has a radius of 6.

Answer ? Answer ?

222 ryx

3622 yx

Example #2: Identify the center and the radius of Example #2: Identify the center and the radius of the following:the following:

Answer ? Answer ?

Center: Center: (0,0) (0,0)

Radius Radius 1010

10022 yx

Write an equation for the following in standard Write an equation for the following in standard form.form.

2522 yx

Graph the Equation of a CircleGraph the Equation of a Circle

Graph xGraph x22 = 25 – y = 25 – y22

Write an Equation of a CircleWrite an Equation of a Circle

The point (6, 2) lies on a circle whose center is The point (6, 2) lies on a circle whose center is the origin. Write the standard form of the the origin. Write the standard form of the equation of the circle.equation of the circle.

We need to find the radius.We need to find the radius. Use the distance formula.Use the distance formula.

212

212 yyxxrd

If a circle has a translated center then the new If a circle has a translated center then the new equation will be:equation will be:

Where the new center will be (h, k) and “r” will be Where the new center will be (h, k) and “r” will be the radius.the radius.

222 )()( rkyhx

Example #1. Write the standard equation for a circle Example #1. Write the standard equation for a circle whose center is at (-3,5) and has a radius of 7.whose center is at (-3,5) and has a radius of 7.

Answer: Since h=-3 and k=5 and r=7 we can substitute Answer: Since h=-3 and k=5 and r=7 we can substitute these into the equationthese into the equation

Therefore:Therefore:

Or: Or:

222 )()( rkyhx

222 7))5(())3(( yx

49)5()3( 22 yx

Example #2 Given the following equation, identify the Example #2 Given the following equation, identify the center and the radius.center and the radius.

Center ? Center ?

Did you say (7,-2)Did you say (7,-2)

Radius ?Radius ?

I bet you said (drum roll please)I bet you said (drum roll please)

99

You guys are AWESOME !!!You guys are AWESOME !!!

81)2()7( 22 yx

Example #3 Given the following graph answer Example #3 Given the following graph answer the following questions.the following questions.

1. Identify the center.1. Identify the center.

2. Name the radius.2. Name the radius.

3. Write an equation in standard form for the 3. Write an equation in standard form for the circle.circle.

AnswersAnswers

Center: (-2,-1)Center: (-2,-1)

Radius: 8Radius: 8

Equation: Equation:

64)1()2( 22 yx

Finding the equation of a Tangent Finding the equation of a Tangent Line to a CircleLine to a Circle

Any line tangent to a circle will be Any line tangent to a circle will be perpendicularperpendicular to a line that goes through the tangent point and to a line that goes through the tangent point and the center of the circle the center of the circle

Perpendicular lines have Perpendicular lines have negative reciprocalnegative reciprocal slopes slopes• (Opposite Reciprocal)(Opposite Reciprocal)

So if we have the center of a circle, and the point of So if we have the center of a circle, and the point of tangency, we can find the slope by the slope formulatangency, we can find the slope by the slope formula

Take the negative reciprocal and use the point slope Take the negative reciprocal and use the point slope formula for a line that goes through the tangent pointformula for a line that goes through the tangent point

run

rise

x

y

xx

yym

12

12

11 xxmyy

Find the equation of a Tangent Line to Find the equation of a Tangent Line to a circlea circle

Find the equation of a tangent line to the circle Find the equation of a tangent line to the circle x x22 + y + y22 = 10 at (-1, 3) = 10 at (-1, 3)

AssignmentAssignment

Pg 182Pg 182

1 – 23 odd, 241 – 23 odd, 24

Pg 183Pg 183

1 – 17 odd, 181 – 17 odd, 18