Introduction to Conics & Circles Chapter 11

1. Introduction to Conics&CirclesChapter 11

2. Conics The conics get their name from the fact that they can be formed by passing a plane through a double-napped cone (two right circular cones placed together, nose-to-nose).

3. Conics Conic sections were studied by the ancient Greeks from a geometric point of view, but today we describe them in terms of the coordinate plane and distance, or as graphs of equations.

4. Analytic Geometry The study of the geometric properties of objects using a coordinate system is called analytic geometry (hence, the title of chapter 11).

5. Typical Conic Shapes Horizontal Parabola Circle Horizontal Hyperbola Vertical Parabola Vertical Ellipse Vertical Hyperbola

6. First conic section: CIRCLES

7. Definition of Circle A circle is the set of all points that are the same distance, r, from a fixed point (h, k). Thus, the standard equation of a circle has been derived from the distance formula.

8. Derive the equation for a circle Given the distance formula, derive the standard equation for a circle. d = d = r =

9. Standard Form of the Circle (h, k) represents the __________ r represents the ___________

10. Example #1 Write an equation of a circle in standard form with a center of (4, 3) and a radius of 5. Then graph the circle.

11. Example #2 Write an equation of a circle in standard form with a center of (2, -1) and a radius of 4. Then graph the circle.

12. Example #3 Write the equation in standard form for the circle centered at (–5, 12) and passing through the point (–2, 8). (x + 5)2 + (y – 12)2 = 25

13. General Form of the Circle x2+ y2 + Ax + By + C = 0

14. Example #4 What is the equation of the circle pictured below? Write the equation in both standard form andgeneral form.

15. Example #5 Graphthecircle. x2+ y2 - 6x + 4y + 9 = 0