Keystone Geometry Unit 7 Equation of A Circle Keystone Geometry Unit 7

Equation of a Circle The center of a circle is given by (h, k). The radius of a circle is given by r. The equation of a circle in standard form is: (x – h)2 + (y – k)2 = r2

Finding the Equation of a Circle Circle O The center is (0, 0) so it is located at the origin. The radius is 12. The equation is x 2 + y 2 = 144

Finding the Equation of a Circle Circle A The center is (16, 10) The radius is 10 The equation is (x – 16)2 + (y – 10)2 = 100 Circle B The center is (4, 20) The radius is 10 (x – 4)2 + (y – 20)2 = 100

Graphing Circles (x – 3)2 + (y – 2)2 = 9 Center (3, 2) Radius of 3

Graphing Circles (x + 4)2 + (y – 1)2 = 25 Center (-4, 1) Radius of 5

Graphing Circles (x – 5)2 + y2 = 36 Center (5, 0) Radius of 6

Writing Equations of Circles Write the standard equation of the circle with a center at (4, 7) and a radius of 5. (x – 4)2 + (y – 7)2 = 25 (-3, 8) and a radius of 6.2 (x + 3)2 + (y – 8)2 = 38.44

Writing Equations of Circles Write the standard equation of the circle with a center at (2, -9) and a radius of . (x – 2)2 + (y + 9)2 = 11 (0, 6) and a radius of . x 2 + (y – 6)2 = 7

Writing Equations of Circles Write the standard equation of the circle with a center at (-1.9, 8.7) and a radius of 3. (x + 1.9)2 + (y – 8.7)2 = 9

Identify the center and radius and sketch the graph: Remember, the center values end up being the opposite sign of what is with the x and y and the right hand side is the radius squared. So the center is at (-4,3) and the radius is 5. 2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 4 6 8

But what if the equation of a circle is not in standard form but multiplied out (FOILED)! Moving everything to one side in descending order and combining like terms we’d have: If we'd have started with it like this, we'd have to complete the square on both the x's and y's to get in standard form.

Move constant to the other side Group x terms and a place to complete the square Group y terms and a place to complete the square 4 16 4 16 Complete the square, write factored form, and we would be back to the standard form for the equation of a circle with a center at (-2,4) and a radius of 2.

Writing a Standard Equation of a Circle The point (1, 2) is on a circle whose center is (5, -1). Write a standard equation of the circle. First solve for the radius: Then plug h, k and r in to the standard equation of a circle:

Another example: Find the center, the length of the radius, and write the equation of the circle if the endpoints of a diameter are (-8,2) and (2,0). Center: Use midpoint formula! Length: use distance formula with radius and an endpoint Equation: Put it all together