c i r c l e

C I R C L EC O N I C S E C T I O N S

(-4, ½ )

Determine the midpoint of the segment whose endpoints are given:

(2, -1)(4, 4)

(4, 0)(5, 4)

1. (5, -3) and (-1, 1)

2. (0, 0) and (8, 8)

3. (10, 1) and (0, 7)

4. (-1, 6) and (9, -6)

5. (-4, -5) and (-4, 6)

THE DISTANCE FORMULA

(x2, y2)

(x1, y1)

D (x2 x1)2 (y2 y1)

2

NextPreviousMain Menu

1

0.5

-0.5

-1

-1.5

-2 -1 1 2

Deriving the Equation of a CircleEnd

A CIRCLE is….

…a SET OF POINTS in a plane…..…each EQUIDISTANT from a fixed point called the CENTER.

A CIRCLE is….

… a set of points (LOCUS) in a plane each EQUIDISTANT from a fixed point called the CENTER.

RADIUS (R)

P R O P E R T I E S

DIAMETER (D)

P R O P E R T I E S

RADIUS (R)

P R O P E R T I E S

DIAMETER (D)CHORDSECANT LINETANGENT LINE

CHORD

P R O P E R T I E S

RADIUS (R)DIAMETER (D)CHORDSECANT LINE

P R O P E R T I E S

The equation of a circle is derived from its radius.

1

0.5

-0.5

-1

-1.5

-2 -1 1 2

NextPreviousMain MenuEnd Last Viewed

Standard Equation of a Circle

Use the distance formula to find an equation for x and y. This equation is also the equation for the circle.

THE DISTANCE FORMULA

(x2, y2)

(x1, y1)

D (x2 x1)2 (y2 y1)

2

NextPreviousMain Menu

1

0.5

-0.5

-1

-1.5

-2 -1 1 2

Deriving the Equation of a CircleEnd

Deriving the Equation of a Circle

D x2 x1 2 y2 y1 2

r x 0 2 y 0 2

r x2 y2

r2 x2 y2 2

1

0.5

-0.5

-1

-1.5

-2 -1 1 2

(x, y)

(0, 0)

Let r for radius length replace D for distance.

r2 = x2 + y2r2 = x2 + y2

Is the equation for a circle with its center at the origin and a radius of length r.

NextPreviousMain MenuEnd

x2 + y2 = 4

If r = 2, then(x, y)

(0, 0)

r = 2

NextPreviousMain MenuDeriving the Equation of a CircleEnd

Q1: What is the equation of a circle if the center is no longer at the origin?

1

0.5

-0.5

-1

-1.5

-2 -1 1

Q2: What changes occur if the equation of the circle is

(x- 3)2 + (y – 1)2 = 4?

(x,y)

Assume (x, y) are the coordinates of a point on a circle.

(h,k)

The center of the circle is (h, k),

r

and the radius is r.

Then the equation of a circle is: (x - h)2 + (y - k)2 = r2.

NextPreviousMain MenuMoving the CenterEnd

6

4

2

-2

-4

-6

-10 -5 5 10

(x - h)2 + (y - k)2 = r2

(x - 0)2 + (y - 3)2 = 72

(x)2 + (y - 3)2 = 49

Write the equation of a circle with a center at (0, 3) and a radius of 7.

NextPreviousMain MenuMoving the CenterEnd

20

15

10

5

-5

-20 -10 10 20

Example 1

Find the equation whose diameter has endpoints of (-5, 2) and (3, 6). First find the midpoint of the diameter using the midpoint formula.

MIDPOINT

M(x,y) x1x2

2,y1y2

2

This will be the center.

M(x,y) 53

2,26

2

M(x,y) 1, 4

NextPreviousMain MenuMoving the CenterEnd

Example 2

Find the equation whose diameter has endpoints of (-5, 2) and (3, 6).

Then find the length distance between the midpoint and one of the endpoints.

DISTANCE FORMULA

This will be the radius.

2122

12 yyxxd

22 6431 radius

22 24 radius416radius

20radiusNextPreviousMain MenuMoving the CenterEnd

Find the equation whose diameter has endpoints of (-5, 2) and (3, 6).

Therefore the center is (-1, 4)

The radius is

20

(x - -1)2 + (y - 4)2 =

202

Ans: (x + 1)2 + (y - 4)2 = 20

NextPreviousMain MenuMoving the CenterSkip TangentsEnd

-10

10

5

-5

Find the center and radius of the circle having the equation x2 + y2 - 10x + 4y + 13 = 0. Graph the circle.

1. Move the CONSTANT term to the other side.

3. Complete the square for the x-terms and y-terms.

x2 - 10x + y2 + 4y = -13

102

5

5 2 25

42

2

2 2 4

x2 - 10x + 25 + y2 + 4y + 4 = -13 + 25 + 4

(x - 5)2 + (y + 2)2 = 16

x2 + y2 - 10x + 4y + 13 = 0

x2 + y2 - 10x + 4y = -13

2. Group the x-terms and y-terms together

x2 - 10x + y2 + 4y = -13

NextPreviousMain MenuCompleting the SquareEndExample 3

(x - 5)2 + (y + 2)2 = 16

center: (5, -2)

radius = 4

(x - 5)2 + (y + 2)2 = 42

4

(5, -2)

NextPreviousMain MenuCompleting the SquareEnd

-5 5 10

6

4

2

-2

-4

-6

HW: On a graphing paper, answer the following:

1. Find the center and radius of the circle whose equation is given by 2x2 +2y2 -2x + 2y - 7 = 0. Draw the sketch.

2. Find the equation of the circle if it has a diameter whose endpoints are at (-1, -5) and (4, -6). Express your answer in the standard and general form.