Circle

Example 1: Give the standard equation of the circle satisfying the given conditions.

Center at the origin, radius 5

Center (4, 3), radius ?9

Center (-2, 6), radius 4

Solution:

Equation of the circle: (x-h)^2+(y-k)^2=r^2

The center is (0, 0) therefore the equation of the circle is x^2+y^2=25.

h=4 and k= 3, the radius is?9, therefore the equation is (x-4)^2+(y-3)^2=9.

h=-2 and k= 6, the radius is 4, therefore the equation is (x+2)^2+(y-6)^2=16.

Example 2: A circle C has a center (2,5) and passes through the point (0, 0). Find the equation for C.

Solution: (x-h)^2+(y-k)^2=r^2

r= (2)^2+(5)^2=29

Therefore the equation of c is: (x-2)^2+(y-5)^2=29.

Example 3: Given the equation, Find the center and radius of the circle. Sketch and indicate the center of the graph.

x^2+y^2-2x=15

x^2+6x+y^2+8=12

Solution: Rewrite the equation in standard form by completing the square in x and y. We can determine the center and the radius using the standard form of the equation.

x^2+y^2-2x=15

x^2+2x+1+y^2=15+1

(x+1)^2+y^2=16

Center (1,0) , r=4

x^2+6x+y^2+8y=12

x^2+6x+9+y^2+8y+16=12+9+16

(x+3)^2+(y+4)^2=37

Center (3,4), r=?37

Example 4: