EXAMPLE 1 Graph an equation of a circle

Graph y2 = – x2 + 36. Identify the radius of the circle.

SOLUTION

STEP 1

Rewrite the equation y2 = – x2 + 36 in standard form as x2 + y2 = 36.

STEP 2

Identify the center and radius. From the equation, the graph is a circle centered at the origin with radius

r = 36 = 6.

EXAMPLE 1 Graph an equation of a circle

STEP 3

Draw the circle. First plot several convenient points that are 6 units from the origin, such as (0, 6), (6, 0), (0, –6), and (–6, 0). Then draw the circle that passes through the points.

EXAMPLE 2 Write an equation of a circle

The point (2, –5) lies on a circle whose center is the origin. Write the standard form of the equation of the circle.

SOLUTION

Because the point (2, –5) lies on the circle, the circle’s radius r must be the distance between the center (0, 0) and (2, –5). Use the distance formula.

r = (2 – 0)2 + (–5 – 0)2 = 29= 4 + 25

The radius is 29

EXAMPLE 2 Write an equation of a circle

Use the standard form with r to write an equation of the circle.

= 29

x2 + y2 = r2 Standard form

x2 + y2 = ( 29 )2 Substitute for r29

x2 + y2 = 29 Simplify

EXAMPLE 3 Standardized Test Practice

SOLUTION

A line tangent to a circle is perpendicular to the radius at the point of tangency. Because the radius to the point (1–3, 2) has slope

= 2 – 0 – 3 – 0 = 2

3 –m

EXAMPLE 3 Standardized Test Practice

23

–the slope of the tangent line at (–3, 2) is the negative reciprocal of or An equation of3

2the tangent line is as follows:

y – 2 = (x – (– 3))32

Point-slope form

32

y – 2 = x + 92

Distributive property

32

13 2

y = x + Solve for y.

ANSWER

The correct answer is C.

GUIDED PRACTICE for Examples 1, 2, and 3

Graph the equation. Identify the radius of the circle.

1. x2 + y2 = 9

SOLUTION 3

GUIDED PRACTICE for Examples 1, 2, and 3

2. y2 = –x2 + 49

SOLUTION 7

GUIDED PRACTICE for Examples 1, 2, and 3

3. x2 – 18 = –y2

SOLUTION 2

GUIDED PRACTICE for Examples 1, 2, and 3

4. Write the standard form of the equation of the circle that passes through (5, –1) and whose center is the origin.

SOLUTION x2 + y2 = 26

5. Write an equation of the line tangent to the circle x2 + y2 = 37 at (6, 1).

y = –6x + 37SOLUTION