Geometry

Circles in the Coordinate Plane

If you graph the equation y = 3x + 5, its graph is a ____ so it is called a ______ equation.

linelinear

For the equation y = 3x + 5, 3 is the ______ of the line and 5 is the __________ of the line.

Just by looking at the equation, you can determine the slope and y-intercept of the line without graphing. For this reason, the equation is said to be written in _____________ form.

slopey-intercept

slope-intercept

Similarly, there is a special form for the graph of a circle. This equation

has the unofficial name of the form of a circle.center- radius

Recall that a circle is the set of points

in a plane, that are equidistant from a given point.

Example 1: Use the distance formula to determine the three distances below. Show your initial use of the distance formula.

CO =

HO =

KO =

2 21 2 1 3 5

2 26 2 6 3 5

2 21 2 7 3 5

In addition to equaling the same distance, what do the three

distance formula have in common?

2 2__ 2 __ 3

Using x and y as coordinates of any point on the circle, what would the distance formula look like for OO?

2 2x h y k r

Squaring both sides we get the center-radius form of a circle.

Center:_______Radius: _______

2 2 2x h y k r

(h , k)r

Example 1: Write an equation for the circle with the given center and radius.

a) center: (4, 3) radius = 7

b) center (5, -2) radius =

2 24 3 49x y

2 25 2 27x y

3 3

Example 2: Find the center and radius of each circle.

a) center ( ____ , ____ ) r = _____

b) center ( ____ , ____ ) r = _____

2 2( 4) ( 3) 43x y

2 25 100x y

3 4 43

0 5 10

Example 3: Write the equation of the circle to the right.

2 2( 2) 16x y

Example 4: Write an equation of a circle with a center of (3, 4) containing the point (8, -8).

2 23 8 4 8 13

22( 3) 4 169x y

Example: The equation of a circle is x2 + y2 – 6x + 2y = b. Find the center of the circle.

If the radius of the circle is 8 units, determine the value of b.

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191296 22 byyxx

1013 22 byx

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6410 b

54b