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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
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Write the standard form of the circleWrite the standard form of the circle
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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
