Equations of Circles

Crop Circles Whether you believe they are fact or faked… they are full of mathematics!!! http://www.youtube.com/watch?v=3bqNvM7op64

Introductory Group Activity Complete the introductory group activity on pages 1 & 2 of your packet

Notes: Writing Equations of Circles Given any point on a circle with center (0,0) the Pythagorean Theorem gives us x2 + y2 = r2 r y x

Example Write the equation for the circle shown in the graph r= Equation = x2 + y2 = r2= 3 9 x2 + y2 = 9

But what if the circle is NOT centered at (0,0)? Suppose a circle is centered at point (h,k). Use the distance formula to find r.

Examples Write the standard equation for a circle with center (-2,5) and radius 7 (x-h)2 + (y-k)2 = r2 (x- -2)2 + (y- 5)2 = 72 (x+2)2 + (y-5)2 = 49

Example B) The point (-5,6) is on a circle with center (-1,3). Write the equation for the circle. **Hint: You have to solve for r first! Then write the equation. (x--1)2 + (y-3)2 = 52 (x-h)2 + (y-k)2 = r2 (-5- -1)2 + (6- 3)2 = r2 (x+1)2 + (y-3)2 = 25 (-4)2 + (3)2 = r2 16+ 9 = r2 25 = r2 5 = r

Graphing Circles Example: Graph the circle with the equation (x-4)2 + (y+2) 2 = 36 Determine center: Radius is: Draw point at center; mark radius up, down, left and right and connect freehand or with compass. (4,-2) Sqrt(36) = 6

Example (-1,2)

Examples Write equations for the circles shown. a) x2 + y2 = 4 b) x2 + y2 = 36 c) (x-2)2 + (y-3)2 = 4

Examples Write the standard equation of the circles with the given center and radii: A) center: (0,0) radius: 3 B) center: (-2,5) radius: 7 a) x2 + y2 = 36 b) (x+2)2 + (y-5)2 = 49

Examples Write the standard equation of a circle with the given center and point on the circle: center: (1,4) point (3,4) center: (2,6) point (-1,2) center: (-1,2) point (-3,4) a) (x-1)2 + (y-4)2 = 4 b) (x-2)2 + (y-6)2 = 25 c) (x+1)2 + (y-2)2 = 8

Examples Graph the circles

Example Burial Location: (-4,8)