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CONIC SECTIONS

CONIC SECTIONS. Quadratic Relations. Parabola. Circle. Ellipse. Hyperbola. The general equation for all conic sections is:. Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0. where A, B, C, D, E and F represent constants.

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CONIC SECTIONS

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  1. CONIC SECTIONS Quadratic Relations Parabola Circle Ellipse Hyperbola

  2. The general equation for all conic sections is: Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 where A, B, C, D, E and F represent constants

  3. The graph of a quadratic relation will be an ellipse if the coefficients of the x2 term and y2 term are different andboth positive(and the xy term is zero).

  4. Definition of an ellipse An ellipse is the set of all points whose distances from two fixed points have a constant sum. d1 + d2 = a constant value.

  5. Foci (plural of focus) The two fixed points in an ellipse. Label these on your index card. (These are your pin holes.) Focus Focus

  6. Vertices The two points at the intersection of the line through the foci and the ellipse. Label these on your index card. Vertex Vertex

  7. Major Axis The line segment joining the vertices. Draw this on your index card. Major Axis

  8. Minor Axis The line segment perpendicular to the major axis at the midpoint. Draw this on your index card. Minor Axis

  9. Co-vertices The endpoints of the minor axis. Label these on your index card. Co-vertex Co-vertex

  10. In a general ellipse with the center at the origin… • a is the distance from the center to the vertex • b is the distance from the center to the co-vertex b a

  11. So the coordinate of the right vertex is (a, 0). The coordinate of the left vertex is (-a, 0). Label your index card. (a,0) (-a,0) a a 2a

  12. The coordinate of the top co-vertex is (0, b) The coordinate of the bottom vertex is (0, -b) Label your index card. (0,b) b 2b b (0,-b)

  13. The foci of the ellipse lie on the major axis, c units from the center, where c2 = a2 - b2 (-c,0) (c,0) c c

  14. The standard form of the equation of an ellipse with center at (0,0) and major and minor axes of length 2a and 2b, where a>b, is as follows. Horizontal major axis Vertical major axis Notice the switch!

  15. What does an ellipse with a vertical major axis look like? Vertex Major axis Focus Co-vertex Co-vertex Minor axis Focus Vertex

  16. What is Eccentricity? Eccentricity tells us how flat (or round) the ellipse is. As e approaches 0, the ellipse becomes a circle. As e approaches 1, the ellipse flattens to a line.

  17. Example 1: Find the vertices and foci of the ellipse a2 = 16 so a = 4 The vertices are (4, 0) and (-4, 0). We know that c2 = a2 – b2. So, c2 = 16 – 7 = 9and c = 3. The foci are (3, 0) and (-3, 0).

  18. Example 2: Find an equation for the ellipse with foci (0, -3) and (0, 3) whose minor axis has length 4. Minor axis = 2b, so b = 2. a2 = b2 + c2 a2 = 22 + 32 a2 = 13 The major axis is vertical, so the a2 goes with the y.

  19. Example 3: Graph a = 8 b = 6

  20. An ellipse with center (h,k)

  21. An ellipse with center (h,k)

  22. Example 4: Find the standard form of the equation for the ellipse whose major axis has endpoints (-2, -1) and (8, -1) and whose minor axis has length 8. (h, k) is the midpoint of major axis → (3, -1) Major axis = 10, so a = 5 Minor axis = 8, so b = 4

  23. Example 5: Find the center, vertices, and foci of the ellipse Center (h, k) = (-2, 5) a = 7 b = 3 Vertices: (h, k ± a) = (-2, 5 ± 7) → (-2, 12) and (-2, -2) Foci: (h, k ± c) = (-2, 5 ± ) c2 = a2 - b2 → c =

  24. HW • 653 Exercise 1-6 and 21-26

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