1.5 – Circles (Lesson Objectives)
• 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.
DefinitionA 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) 2 2 2x h y k r
r is the radius of the circle
Every binomial squared has been multiplied out.
Every term is on the left side, set equal to 0.
2 2: 4 6 3 0 Example x y x y
General Form of a Circle
EX 1 Write the standard form for the equation of a circle with center (3, -2) and a radius of 4.
2 2 2+x h y k r
22 23 + 2 4x y
2 23 + 2 16x y
EX 2 Write an equation of a circle with center (-4, 0) and a diameter of 10.
2 2 2+x h y k r
2 2 24 + 0 5x y
2 24 +y 25x
EX 3 Write an equation of a circle with center (2, -9) and a radius of .
11
2222 9 11x y
2 2 2+x h y k r
2 22 + y+9 11x
EX 4 Find the coordinates of the center and the measure of the radius.
2 2 26 + 3 25x y
The center is
The radius is
The equation is
5. Find the center, radius, & equation of the circle.
(0, 0)
12
x2 + y2 = 144
The center is
The radius is
The equation is
6. Find the center, radius, & equation of the circle.
(1, -3)
7
(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.
2 2( 8 ) 7 x x y
8. Write the standard equation of the circle. State the center & radius.
2 2 8 7 0x y x
Center: (4, 0) radius: 3
2 2( 4) 9x y
2 2( 8 )16 167 x x y
2 24 6 3 x x y y
9. Write the standard equation of the circle. State the center & radius.
2 2 4 6 3 0 x y x y
Center: (-2, 3) radius: 4
2 22 3 16 x y
2 24 6 9394 4 x x y y
2 28 2 10 x x y y
10. Write the standard equation of the circle. State the center & radius.
2 22 2 16 4 20 0 x y x y
2 24 1 7 x y
2 28 2 1 116 1610 x x y y
2 2 8 2 10 0 x y x y
: (4, 1) : 7 2.6 Center Radius
11. Write the general form of the equation of the circle.
2 24 3 36 x y
2 28 16 6 9 36 x x y y2 28 16 6 9 36 0 x x y y
2 2 8 6 11 0 x y x y
Homework• 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