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CirclesCircles
5.3 (M3)5.3 (M3)
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
EXAMPLE 4 Write a circular model
Cell Phones
A cellular phone tower services a 10 mile radius. You get a flat tire 4 miles east and 9 miles north of the tower. Are you in the tower’s range?
SOLUTION
STEP 1
Write an inequality for the region covered by the tower. From the diagram, this region is all points that satisfy the following inequality:
x2 + y2 < 102
In the diagram above, the origin represents the tower and the positive y-axis represents north.
EXAMPLE 4 Write a circular model
STEP 2
Substitute the coordinates (4, 9) into the inequality from Step 1.
x2 + y2 < 102 Inequality from Step 1
42 + 92 < 102? Substitute for x and y.
The inequality is true.97 < 100
ANSWER
So, you are in the tower’s range.
Write the standard form of the circleWrite the standard form of the circle
Complete the square to create the Complete the square to create the standard formstandard form 2 2 2( ) ( )x h y k r
2 2 4 6 3 0x y x y
2 2( 2) ( 3) 16x y
Write the standard form of the circleWrite the standard form of the circle
2 2 2 2 7 0x y x y
2 2( 1) ( 1) 9x y
Write the standard form of the circleWrite the standard form of the circle
2 2 8 7 0x y x
2 2( 4) 9x y
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