10-7 Circles in the Coordinate Plane The equation of a circle is based on the Distance Formula and the fact that all points on a circle are equidistant from the center.

NOTE: The “sign” of h and k flips between the equation and the coordinate point (h, k)

Example 1A Write the equation of each circle. Q with r = 4 and has center Q(2, –1) (x – h)2 + (y – k)2 = r2 Equation of a circle (x – 2)2 + (y + 1)2 = 16 Write equation of the circle

Example 1B: Writing the Equation of a Circle Write the equation of each circle. K that passes through J(6, 4) and has center K(1, –8) (x – h)2 + (y – k)2 = r2 Equation of a circle   Find r Write equation of the circle (x – 1)2 + (y + 8)2 = 169

Example 2A: Graphing a Circle Graph (x – 3)2 + (y + 4)2 = 9. The equation of the given circle can be written as (x – 3)2 + (y – (– 4))2 = 32. (3, –4) So h = 3, k = –4, and r = 3. The center is (3, –4) and the radius is 3. Plot the point (3, –4). Then graph a circle having this center and radius 3.

Example 2B Graph (x – 3)2 + (y + 2)2 = 4. Center: (3, -2) Radius: 2 (3, –2) Radius: 2 Graph

IS A GIVEN POINT “ON”, “INSIDE” OR “OUTSIDE” THE CIRCLE If distance to point = r then “ON” the circle If distance to point < r then “INSIDE” the circle If distance to point is > r then “OUTSIDE” the circle

  INSIDE   ON

Lesson Quiz: Part I Write the equation of each circle. 1. L with center L (–5, –6) and radius 9 2. D that passes through (–2, –1) and has center D(2, –4)

Lesson Quiz: Part II Graph each equation. 3. x2 + y2 = 4 4. (x – 2)2 + (y + 4)2 = 16

Honors Lesson Quiz: Part III An amateur radio operator wants to build a radio antenna near his home without using his house as a bracing point. He uses three poles to brace the antenna. The poles are to be inserted in the ground at three points equidistant from the antenna located at J(4, 4), K(–3, –1), and L(2, –8). What are the coordinates of the base of the antenna?

Completing the Square Sometimes it is necessary to complete the square to write the equation of a circle in the form of: (x – h)2 + (y – k)2 = r2 Steps: 1. Divide the equation by a constant if necessary to make the coefficients of the squared terms = 1 2. Group the x terms and y terms, move the constant terms to the other side, leave blanks. 3. Take ½ the 1st linear coefficient and square it; add that number to both sides filling in the 1st blank. Repeat for the second linear term and fill in the second blank. 4. Factor the left side.

Example 1 1.  by 2 → x2 + y2 – 4x + 12y + 31 =0 2. x2– 4x + __ + y2 + 12y + __ = -31 + __ + __ 3. x2– 4x + _4 + y2 + 12y + 36 = -31 + 4_ + 36 4. (x – 2) 2 + (y + 6) 2 = 9

Example 2 3x2 + 3y2 + 6x – 18y = 15

Example 3 4x2 + 4y2 – 16y = 15