Download 1.5 - equations of circles

1.5 – Circles (Lesson Objectives)
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Write the standard form of the
equation of a circle.
Graph a circle by hand and with a
calculator using the standard form of
the equation of a circle.
Work with the general form of an
equation of a circle.
Definition
A Circle is a set of
points in the xy
plane that are a
fixed distance r from
a fixed point (h,k),
where (h,k) is the
center of the circle
and r is the radius.
Standard Form of a Circle
Center is at (h, k)
 x  h
2
 y  k  r
2
2
r is the radius of the circle
General Form of a Circle
Every binomial squared has been
multiplied out.
Every term is on the left side, set
equal to 0.
Example :
x  y  4x  6 y  3  0
2
2
EX 1 Write the standard form for the equation of a
circle with center (3, -2) and a radius of 4.
 x  h +  y  k 
2
2
 x  3  +  y   2  
2
 x  3 + y  2
2
2
r
2
2
4
2
 16
EX 2 Write an equation of a circle with
center (-4, 0) and a diameter of 10.
 x  h +  y  k 
2
2
r
2
 x   4  +  y  0  5
2
2
 x  4  +y  25
2
2
2
EX 3 Write an equation of a circle with
center (2, -9) and a radius of 11.
 x  h +  y  k 
2
 x  2
2
2
r
2
  y   9   11
2
2
 x  2  +  y+9
2
2
 11
EX 4 Find the coordinates of the center and
the measure of the radius.
 x  6 +  y  3
2
2
 25
2
5. Find the center, radius, &
equation of the circle.
The center is (0, 0)
The radius is 12
The equation is
x2 + y2 = 144
6. Find the center, radius, &
equation of the circle.
The center is (1, -3)
The radius is 7
The equation is
(x – 1)2 + (y + 3)2 = 49
7. Graph the circle, identify
the center & radius.
(x – 3)2 + (y – 2)2 = 9
Center (3, 2)
Radius of 3
Converting from General Form to
Standard
1. Move the x terms together and the
y terms together.
2. Move C to the other side.
3. Complete the square (as needed)
for x.
4. Complete the square(as needed)
for y.
5. Factor the left & simplify the right.
8. Write the standard equation of the circle.
State the center & radius.
x  y  8x  7  0
2
2
( x  8x
)  y  7
2
2
( x  8 x  16)  y  7  16
2
2
( x  4)  y  9
2
2
Center: (4, 0) radius: 3
9. Write the standard equation of the circle.
State the center & radius.
x  y  4x  6 y  3  0
2
x
x
2
2
 4x
2
y
2
 6y
3
 4x  4   y  6 y  9  3  4  9
2
2
2
 x  2    y  3
 16
Center: (-2, 3) radius: 4
10. Write the standard equation of the circle.
State the center & radius.
2 x  2 y  16 x  4 y  20  0
2
2
x  y  8 x  2 y  10  0
2
x
x
2
2
2
y
 8x
2
 2y
  10
 8 x  16    y  2 y  1  10  16  1
 x  4
2
2
  y  1  7
2
Center : (4, 1) Radius : 7  2.6
11. Write the general form of the equation of
the circle.
2
2
 x  4
  y  3  36
x  8 x  16  y  6 y  9  36
2
2
x  8 x  16  y  6 y  9  36  0
2
2
x  y  8 x  6 y  11  0
2
2
Homework
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1.5 - p.49 1-4, 7-43 by
4’s, 45
Bring in textbook on
Monday
Bring in your Notecard
for Tuesday’s Ch.1 Test