9.3 Graph and Write Equations of Circles

Circles circle: in a plane, the set of points equidistant from a given point, called the center. radius: any segment whose endpoints are the center and a point on the circle. If the circle is centered at (0, 0), and the radius is r, then the distance to any point, (x, y) on the circle (using the distance formula) is (x,y) square both sides... now simplify... This is the standard form of a circle with center (0,0) and radius r.

Example: Write an equation of the circle with its center at the origin with the point (–3, 4) on the circle. Use the Standard Equation of the Circle to find the radius of the circle...

Draw picture, what does tangent mean? Find the slope from the center to the point (2,3) The tangent line is perpendicular to this radius so the slope must be the negative reciprocal

Use substitution… x2 + (2/3 x + 2)2 = 25 Example: Find the points of intersection, if any, of the graphs of x2 + y2 = 25 and y = 2/3 x + 2 Use substitution… x2 + (2/3 x + 2)2 = 25 simplify ... x2 + 4/9 x2 + 8/3 x + 4 = 25 13/9 x2 + 8/3 x + 4 = 25 13/9 x2 + 8/3 x – 21 = 0 multiply by 9 13x2 + 24x – 189 = 0 factor (13x + 63)(x – 3) = 0 13x + 63 = 0 or x – 3 = 0 x = –63 or x = 3 13 Now, find the y-coordinate for each x-coordinate... y = 2/3(–63/13) + 2 y = 2/3 (3) + 2 y = –126/39 + 2 y = 2 + 2 y = –126/39 + 78/39 y = 4 y = –48/39 y = –16/13 The points of intersection are (–63/13, –16/13) and (3, 4)

All points INSIDE the circle. All points OUTSIDE the circle