1. WARM UP 13 (3/2, -5/2) Find the distance between the points 1. (8, 7) and (3, -5)2. (-5, 3) and (2, -7) Find the coordinates of the midpoints of the segments having the following endpoints. 3. (-2, 1) and (5, -6)4. (9, -6) and (9, -4) √149 (9, -5)

2. CONIC SECTIONS: CIRCLES

3. OBJECTIVES Find the equation of a circle given the radius and the coordinates of the center of the circle. Find the radius and the coordinates of the center of a circle given the equation of the circle Use the technique of completing the square to find the radius and coordinates of a circle

4. KEY TERMS & CONCEPTS Center of a circle Circle Cone Conic section Element of a cone Radius of a circle Vertex of a cone

5. CONIC SECTIONS A cone is formed by rotating one of the two distinct, non-perpendicular, intersecting lines about the other. The point of intersection is the vertex of the cone. Any position of the rotating line is an element of the cone. The intersection of any plane with a cone is called a conic section.

6. EXAMPLES Some conic sections are shown below: ConeEllipseParabolaHyperbola

7. EQUATIONS OF CIRCLES Some equations of second degree have graphs that are circles. DEFINITION A circle is the set of all points in a plane that are at a constant distance, the radius, from a fixed point in that plane. The fixed point is the center of the circle. We first consider an equation for a circle centered at the origin. Theorem 10-3 The equation in standard form, of the circle centered at the origin with radius r is

8. EXAMPLE 1 Find an equation of a circle with center at (0, 0) and radius √5. Standard form for a circle centered at the origin. Substituting √5 for r

9. EQUATION OF A CIRCLE Consider an equation for a circle with its center anywhere in the coordinate plane. Theorem 10-4 The equation in standard form, of a circle with center (h, k) and radius r is

10. EXAMPLE 2 Find an equation of a circle with center (-1, 3) and radius √2. Standard form

11. TRY THIS… Find an equation of a circle with center and radius as given a.(0, 0), radius √6 b.(-3, 7), radius √5 c.(5, -2), radius √3 d.(-2, -6), radius 2√7

12. FINDING THE CENTER AND RADIUS Find the center and radius of (x – 2) + (y + 3) = 16 We first write standard form. The center is (2, -3) and the radius is 4. We then draw the graph using a compass. (2, -3)

13. TRY THIS… Find the center and radius of (x + 1) + (y – 3) = 4. Then graph the circle

14. STANDARD FORM BY COMPLETING THE SQUARE Completing the square allows us to find the standard form for the equation of a circle. Example 4: Find the center and radius of We complete the squares to obtain standard form. (x + 8x ) + (y – 2y ) = -15 In standard form, the equation of the circle is. The center is (-4, 1) and the radius is √2. (x + 8x + 16) + (y – 2y ) = -15 + 16) Completing the square. (x + 8x + 16) + (y – 2y + 1) = -15 + 16 + 1)

15. TRY THIS… a.Find the center and radius of the circle x + y – 14x + 4y – 11 = 0 b.Find the center and radius of the circle x + y – 12 x – 8y + 27 = 0

16. CH. 10.2 HOMEWORK Textbook pg. 436 #2, 4, 8, 14, 18 & 22