1. Equations of Circles

2. Circles (Lesson Objectives)
Write the standard form of the equation of a circle.
Graph a circle by hand using the standard form of the equation of a circle.
Write the general form of an equation of a circle using center and radius and identify center and radius from the general form of equation of a circle.

3. Standard Form of a Circle Center is at (h, k)and radius r
r is the radius of the circle

4. EX 1 Write the standard form for the equation of a circle with center (3, -2) and a radius of 4. Draw it. h k r

5. EX 2 Write an equation of a circle with center (-4, 0) and a diameter of 10.k

6. EX 3 Write an equation of a circle with center (2, -9) and a radius of .k r

7. Opposite signs! ( , ) 6 -3 Take the square root! Radius 25
EX 4 Find the coordinates of the center and the measure of the radius.
Opposite signs!
( , ) 6 -3
Take the square root!
Radius 25

8. 5. Find the center, radius, & equation of the circle.
The center is
The radius is
The equation is
(0, 0) 12
x2 + y2 = 144

9. 7. Graph the circle, identify the center & radius.
(x – 3)2 + (y – 2)2 = 9
Center (3, 2)
Radius of 3

10. General Form of a Circle
General form of equation of a circle is
X2 + Y2+2gx+2fy+c = 0
In this equation center is (-g, -f)
Radius is √f2+g2 – c
Also in general form of a circle, the coefficients of x² and y² has to be same, otherwise it is not a circle equation.
To calculate center and radius, make sure you reduce the coefficients to 1 by dividing by the coefficients of x² and y²

11. Example: What is center and radius of x²+y²+4x+6y-2 = 0
2g = 4 g=2 2f=6 f=3
In this problem g = 2 f = 3 and c = -2
So center is (-2, -3) = (-2, -3) and
radius = √(2)²+(3)²-(-2)  = √13+2 = √15
Another example
x²+y²-8x-4y-5=0 2g=-8 g=-4 2f=-4 f=-2 c=-5
Center = (-(-4), -(-2)) = (4, 2)
and radius = √(-4)²+(-2)²-(-5) = √25 = 5

12. An Example on general form with coefficients not equal to 1
4x²+4y²-36x-8y-48=0
Divide by 4 on both sides
x²+y²-9x-2y-12 = 0
2g=-9 2f=-2
g=-9/2 and f= -1 and c=-12
now center is (9/2, 1)  Radius = √(-9/2)²+(-1)²-(-12) = √33.25 = 5.77

13. Another example Determine equation of a circle with center at (0,1) and which passes through (2,3)
Radius = Distance from center to point (2,3) =√(0-2)²+(1-3)² = √8
So equation is (x-0)²+(y-1)²=(√8)²
                       x²+(y-1)²=8
We can convert it to general form too as follows
                       x²+(y-1)(y-1) = 8
                        x²+y²-2y+1=8
                         x²+y²-2y-7=0