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Conic Sections

Conic Sections. An Introduction. Conic Sections - Introduction. A conic is a shape generated by intersecting two lines at a point and rotating one line around the other while keeping the angle between the lines constant. Conic Sections - Introduction.

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Conic Sections

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  1. Conic Sections An Introduction

  2. Conic Sections - Introduction • A conic is a shape generated by intersecting two lines at a point and rotating one line around the other while keeping the angle between the lines constant.

  3. Conic Sections - Introduction • The resulting collection of points is called a right circular cone. The two parts of the cone intersecting at the vertex are called nappes. Vertex Nappe

  4. Conic Sections - Introduction • A “conic” or conic section is the intersection of a plane with the cone. • The plane can intersect the cone at the vertex resulting in a point.

  5. Conic Sections - Introduction • The plane can intersect the cone perpendicular to the axis resulting in a circle.

  6. Conic Sections - Introduction • The plane can intersect one nappe of the cone at an angle to the axis resulting in an ellipse.

  7. Conic Sections - Introduction • The plane can intersect one nappe of the cone at an angle to the axis resulting in a parabola.

  8. Conic Sections - Introduction • The plane can intersect two nappes of the cone resulting in a hyperbola.

  9. Graphing Circles, Ellipses, Hyperbolas Graph each equation. Describe the graph, find the lines of symmetry, x and y intercepts, domain, and range. 1. 2. 3.

  10. Without graphing, which conic section is this?

  11. What is a circle? • A circle is the set of points equally distant from one central point. • The central point is called the center. • • Center

  12. What does r represent? • The distance from the center to the curve of the circle is called the radius. r

  13. What does d represent? • The diameter is the distance across the circle. d

  14. (x,y) • • Assume that (x,y) are the coordinates of a point on the circle. • Use the distance formula to find the radius. • (h,k)

  15. Equation of a circle • r2=(x - h)2 + (y – k)2 • Let’s investigate!

  16. Example #1 • Find the equation of a circle whose center is at (2, -4) and the radius is 5. • Let’s check our answer.

  17. Example #2 • Write an equation of a circle if the endpoints of a diameter are at (5,4) and (-2, -6). Hint: Draw a picture, then find the center and radius.

  18. Example #3 • Find the center and radius of the circle with equation x2 + y2 = 25. • Graph the circle.

  19. Example #4 • Find the center and radius of the circle with equation x2 + y2 – 4x + 8y – 5 = 0.

  20. On your own… • Find center and radius and graph: x2 + y 2 -10x +8y = -40 2. Write an equation for a circle that passes through the point (-1, 4) with a center at (-3, 6).

  21. Assignment: 10.1 p. 550 #1, 3, 5, 17-28, 62 10.3 p. 564 #27-31 odd, 35, 43, 45, 47, 49, 61, 63, 73, (78 graph)

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