8.3 Graph and Write equations of Circles

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8.3 Graph and Write equations of Circles

• What is the standard form equation for a circle?
• Why do you use the distance formula when writing
  the equation of a circle?
• What general equation of a circle is used when
  the center of the circle is translated?
• How do you put an equation into standard form?

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Standard Equation of a Circle
OP r
P(x,y)
r
O
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Standard Equation of a Circle
An equation for the circle with its center at
(0,0) and a radius of r is
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Graph an equation of a circle
Graph y 2 x 2 36. Identify the radius of
the circle.
SOLUTION
STEP 1
Rewrite the equation y 2 x 2 36 in
standard form as x 2 y 2 36.
STEP 2
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.
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Graph the equation. Identify the radius of the
circle.
1.
x 2 y 2 9
SOLUTION
Equation is in the standard form x 2 y 2
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STEP 1
Identify the Center and radius form the equation,
the graph is a circle centered at the origin with
radius
STEP 2
Draw the circle. First plot several convenient
points that are 3 units from the origin, such as
(0, 3), (3, 0), (0, 3), and (3, 0). Then draw
the circle that passes through the points.
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SOLUTION
STEP 1
Rewrite the equation y 2 x2 49 in standard
form as x 2 y 2 49.
STEP 2
STEP 3
Draw the circle. First plot several convenient
points that are 7 units from the origin, such as
(0, 7), (7, 0), (0, 7), and (7, 0). Then draw
the circle that passes through the points.
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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
circles radius r must be the distance between
the center (0, 0) and (2, 5). Use the distance
formula.
Standard form
x 2 y 2 r 2
x 2 y 2
Simplify
x 2 y 2 29
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4. Write the standard form of the equation of the
circle that passes through (5, 1) and whose
center is the origin.
SOLUTION
Because the point (5, 1) lies on the circle, the
circles radius r must be the distance between
the center (0, 0) and (5, 1). Use the distance
formula.
r (5 0)2 (1 0)2
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x 2 y 2 r 2
Standard form
 
x 2 y 2
 
x 2 y 2 26
Simplify
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SOLUTION
m
the slope of the tangent line at (23, 2) is the
negative reciprocal of or An
equation of
2

3
the tangent line is as follows
Point-slope form
Distributive property
Solve for y.
The correct answer is C.
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5. Write an equation of the line tangent to the
circle x 2 y 2 37 at (6, 1).
SOLUTION
A line tangent to a circle is perpendicular to
the radius at the point of tangency. Because the
radius to the point (6, 1) has slope
the tangent line is as follows
Point-slope form
y 1 6(x 6)
Distributive property
y 1 6x 36
Solve for y.
y 6x 37
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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 towers 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
x 2 y 2 lt 102
STEP 2
Substitute the coordinates (4, 9) into the
inequality from Step 1.
x 2 y 2 lt 102
Inequality from Step 1
So, you are in the towers range.
Substitute for x and y.
The inequality is true.
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Standard Equation of a Circle
The standard equation for a translated circle is
(x h)2 (y k)2 r2 center (h, k) radius r
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Example 2
Write the standard equation of the circle graphed
below.
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Practice
Write the standard equation of a circle with the
following center and radius.
1) C(0,0) radius 9
2) C(2,3) radius 5
3) C(-5,2) radius 4
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Practice
Graph each equation. Label the center and
radius.
1) x2 y2 25
2) (x 2)2 y2 4
3) (x 4)2 (y 3)2 49
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Example 1
Write the standard equation for the circle given
by x2 y2 12x 2y - 8 0. State the
coordinates of its center and give its radius.
(6,1)
Center
Radius
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Example
Write the standard equation for the circle given
by x2 y2 6x 4y - 3 0. State the
coordinates of its center and give its radius.
Then sketch the graph.
(-3,2)
Center
Radius
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Practice
Write the standard equation for the circle given
by x2 y2 - 2x 2y - 7 0. State the
coordinates of its center and give its radius.
Then sketch the graph.
 
1
1
1
Complete the square
1
 
Factor
Center is at (1,-1), r 3
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8.3 Assignment
p. 505, 4-18 even, 22- 28 even, 54, 56 p. 531,
4, 8, 13, 14