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.

Write an equation of a circle 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 = 4 + 25 = 29 The radius is 29

Write an equation of a circle 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 = ( 29 )2 x2 + y2 Substitute for r 29 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

Standardized Test Practice EXAMPLE 3 Standardized Test Practice the slope of the tangent line at (–3, 2) is the negative reciprocal of or An equation of 2 3 2 – 3 the tangent line is as follows: y – 2 = (x – (– 3)) 3 2 Point-slope form 3 2 y – 2 = x + 9 Distributive property 3 2 13 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). SOLUTION y = –6x + 37