Warm upO Find the coordinates of the midpoint of

the segment that has endpoints at (- 5, 4) and (7, -2).

O Find the distance between points at (10, 7) and (- 4, 8)

Lesson 10-2 Circles

Objective: To use and determine the standard and general forms of the

equation of a circleTo graph circles

Conic Sections

Formed when a plane intersects a double right cone.

Degenerate ConicsO formed when the plane passes thru

the vertex of the double right cone.

degenerate

ellipse

degenerate

hyperbola

degenerate

parabola

CircleO a set of points in a plane which are

equidistant from a given point. (the center)

O Radius – the distance from the center to any point on the circle.

CircleOThe Standard Form of a circle with a center at

(0,0) and a radius, r, is……..

OFrom Pythagorean Theorem

222 ryx

                                                                    

center (0,0)radius = 2

Copyright ©1999-2004 Oswego City School District Regents Exam Prep Center

Circles

Now if the center moves off of the origin to point (h, k) we can use the distance formula to find the radius.

(x, y)

(h, k)

22 )()( kyhxr

222 )()( kyhxr

CirclesOThe Standard Form of a circle with a center at

(h,k) and a radius, r, is……..

222 )()( rkyhx

                                                                  

                        

                                                                              

center (3,3)radius = 2

Copyright ©1999-2004 Oswego City School District Regents Exam Prep Center

Example 1OWrite the standard form of the

equation of the circle that is tangent to the x-axis and has its center at (-5, 4). Then graph the equation.

Example 2OThe equation of a circle is:

Write the standard form of the equation. divide by 4

complete the

square

51162444 22 yxyx

4

51___)4(___)6( 22

yyxx4

514622

yxyx

494

51)44()96( 22

yyxx

4

1)2()3( 22 yx

Example 2 cont’dO Therefore the center is (3, -2) and the radius

is

Example 3O Write the standard form of the equation of

the circle that passes through the points at (1,1), (1, 2), and (2, 3). Then identify the center and radius of the circle.

O Use the general equationO Use each point as x and y to create 3 new

equations and solve the system for D, E and F.

O Once found substitute D, E and F back into the original equation and complete the square (twice) to create the equation of the circle.

022 FEyDxyx