Jeff Bivin -- LZHS Precalculus – 2015. Jeff Bivin -- LZHS y x -4 4 4 8 The set of all points satisfying the equation gives the circle with center (h,k)

Jeff Bivin -- LZHS

Precalculus – 2015

Jeff Bivin -- LZHS

y

x

-4 4

4

8

2 2 2( ) ( )x h y k r

The set of all points satisfying the equation gives the circle with center (h,k) and radius = r .

Base = x-h

height = y-k

h

kPythagorean Theorem?

r

Jeff Bivin -- LZHS

Circle

The set of all co-planar points equidistant from a fixed point (center).

Jeff Bivin -- LZHS

CircleEquation: (x – h)2 + (y – k)2 = r2

Jeff Bivin -- LZHS

Circle

x – 3 = 0

y – 5 = 0

x = 3

y = 5

r =r = 11

(x – 3)2 + (y – 5)2 = 121

Jeff Bivin -- LZHS

Graph the following circle9x2 + 36x + 9y2 - 18y - 10 = 89

9x2 + 36x + 9y2 - 18y = 89 + 10

(x + 2)2 + (y - 1)2 = 16

9x2 + 36x + 9y2 - 18y = 99

9

(x2 + 4x + 22 ) + (y2 - 2y + (-1)2 ) = 11 + 4 + 1

Jeff Bivin -- LZHS

Circle (x + 2)2 + (y – 1)2 = 16

x + 2 = 0

y – 1 = 0

x = -2

y = 1

r =r = 4

Jeff Bivin -- LZHS

2

21

4

252

2

5

(x2 + 5x + ) + (y2 + 4y + 22 ) = + + 4

Graph the following circle2x2 + 10x + 2y2 + 8y + 4 = 25

2x2 + 10x + 2y2 + 8y = 25 - 4

(x + )2 + (y + 2)2 =

2x2 + 10x + 2y2 + 8y = 21

2

2

5

4

83

Jeff Bivin -- LZHS

Circle (x + )2 + (y + 2)2 =

x + = 0

y + 2 = 0

x =

y = -2

r =

r =

2

5

4

83

2

5

2

5

2,2

5

2,2

5

4

83

2

832

83

Jeff Bivin -- LZHS

Circles• Find the equation of a circle where the

endpoints of a diameter are

(-5,-7) and(-5,11)

Jeff Bivin -- LZHS

Circles

• Find the equation of a circle where the endpoints of a diameter are

(-4,-6) and(2,2)

Jeff Bivin -- LZHS

Circles

• Find the standard form of the equation of a circle that contains the points (9,4), (1,14) and (9,14).

• Use the idea that the perpendicular bisector of any chord of a circle passes through the center of the circle. Since the given points are on the circle, any pair of the three points are the endpoints of a line segment that is a chord of the circle.

• Find the perpendicular bisector of 2 of the chords. Their intersection will be the center of the circle. Distance from center to any of the points = radius.

2 25 9 41x y

Jeff Bivin -- LZHS

Homework:

Circles Worksheet