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Ellipses

Ellipses. Pre Calculus. What You Should Learn. Write equations of ellipses in standard form and graph ellipses. Given equations of ellipses, find key features. Find eccentricities of ellipses. Plan for today…. Review of Homework Quick Review Little Quiz – Lines Ellipses

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Ellipses

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  1. Ellipses Pre Calculus

  2. What You Should Learn • Write equations of ellipses in standard form and graph ellipses. • Given equations of ellipses, find key features. • Find eccentricities of ellipses.

  3. Plan for today… • Review of Homework • Quick Review • Little Quiz – Lines • Ellipses • Homework • Page 722 # 1-6 all (matching), 7, 11,15, 21, 39 – 47 odd

  4. So Far • Double napped cone • Intersection of plane and the cone created 4 basic sections

  5. Lines Inclination If a non horizontal line has an inclination of θ and slope m, then tan θ = m. Remember if the slope, m, is negative, find θ and add 180o Angle between two linesIf two non perpendicular lines have m1 and m2, the angle between the two lines is: Distance between the point (x1, y1) and a lineAx + By + C = 0 is:

  6. Recognizing a Conic Ax2 + Cy2 + Dx + Ey + F = 0 AC = 0 A or C is zero - no x2 or no y2 term Parabola A = C A is equal to C, same value Circle AC > 0 A and C have the same sign but have different values Ellipse AC < 0 A and C have different signs Hyperbola

  7. Circles Standard form of a circle with the center at (0, 0) and r is the radius: x2 + y2 = r2 Standard form of a circle with the center at (h, k) and radius of r: (x – h)2 + (y – k)2 = r2

  8. Parabolas • Orientation • Vertex (h, k) • Directrix • Focus • Axis of symmetry • “p” • Equations:

  9. You should be able to… • Identify key features of lines • Recognize a Conic • Given an equation of a Circle • Find the center and radius • Given key information about a circle, find the equation in standard form • Given an equation of a Parabola • Find the orientation, focus, vertex, directrix • Sketch • Given key information about a parabola, find the equation in standard form

  10. Quiz Time

  11. Ellipse d1 + d2 = constant d1 d2 focus focus major axis vertex vertex center minor axis Anellipseis the set of all points in the plane for which the sum of the distances to two fixed points (called foci) is a positiveconstant. The major axis is the line segment passing through the foci with endpoints (called vertices) on the ellipse. The midpoint of the major axis is the center of the ellipse. The minor axis is the line segment perpendicular to the major axis passing through the center of the ellipse with endpoints on the ellipse. The sum of the distance from any point on the ellipse to each of the foci remains constant.

  12. Key Identifiers of an Ellipse “a” is the distance from the center to the end of the major axis. Therefore the entire length of the major axis is 2a. The center is the midpoint of the major axis. “a” is always the largest number. “b” is the distance from the center to the end of the minor axis. Therefore the entire length of the minor axis is 2b. The center is the midpoint of the minor axis. “c” is the distance from the center to one of the two focus points. The center is the midpoint of the two focus points (foci). c2 = a2 – b2

  13. y (0, b) a b x a c (0, 0) (–c, 0) (c, 0) (–a, 0) (a, 0) (0, –b) The standard form for the equation of an ellipse with center atthe origin and a major axis that is horizontal is: , with: vertices: (–a, 0), (a, 0) and foci:(–c, 0), (c, 0) where c2 = a2 – b2

  14. y (0, a) (0, c) a c (0, 0) x b b (–b, 0) (b, 0) a (0, -c) (0, –a) The standard form for the equation of an ellipse with center at the origin and a major axis that is vertical is: , with: vertices: (0, –a), (0, a) and where c2 = a2 – b2 foci:(0, –c), (0, c)

  15. (0, 3) y 5 3 (0, 5) 4 (4, 0) x (0, –3) (–4, 0) (0, –5) Example: Sketch the ellipse with equation 25x2 + 16y2 = 400 and find the vertices and foci. 1. Put the equation into standard form. divide by 400 So, a = 5 and b = 4. 2. Since the denominator of the y2-term is larger, the major axisis vertical. 3. Vertices:(0, –5), (0, 5) 4. The minor axisis horizontal andintersects the ellipse at (–4, 0) and (4, 0). 5. Foci:c2 = a2 – b2 (5)2 – (4)2 = 9c = 3 foci:(0, –3), (0,3)

  16. a c (h, k + b) (h – c, k) (h, k –b) (h –a, k) (h + a, k) (h + c, k) The standard form for the equation of an ellipse with center atthe (h, k) and a major axis that is horizontal is: vertices: (h – a, k), (h + a, k) and foci:(h – c, k), (h + c, k) where c2 = a2 – b2 b a (h, k)

  17. (h, k + a) (h, k + c) (h, k -c) (h , k - a) The standard form for the equation of an ellipse with center atthe (h, k) and a major axis that is vertical is: vertices: (h , k – a), (h, k + a) and foci:(h, k – c), (h, k + c) where c2 = a2 – b2 a c (h, k) b (h –b, k) b (h + b, k) a

  18. Note on Orientation To recognize the difference between a vertically oriented and horizontally oriented ellipse: • Put the equation is standard form • If the largest number is under the x2 term – it is horizontal • If the largest number is under the y2 term – it is vertical Remember “a” is always the largest

  19. Example: Find the standard form of the equation of the ellipse having foci at (0,1), and (4, 1) and a major axis length of 6. 1. Identify what information you have. Since the foci occur at (0, 1) and (4, 1) the center of the of the ellipse is (2, 1) - midpoint of the foci The distance from the center to a focus point is 2, c = 2 The length of the major axis = 2a so a =3 2. Calculate b: a2 – b2 = c2 a2 – c2 = b232 – 22 = b2 5 = b2 3. Choose the equation based upon orientation of the ellipse, since a is larger, the major axisis horizontal. Or, since the foci are on a horizontal line, it is horizontal. 4. Put information into the equation.

  20. Example: Sketch a graph of the ellipse: x2 + 4y2 + 6x – 8y + 9 = 0 • Put the equation into standard form by completing the square. 2. Since the denominator of the x2-term is larger, the major axisis horizontal.

  21. Example: Sketch a graph of the ellipse: x2 + 4y2 + 6x – 8y + 9 = 0 Horizontal 3. To graph this we need to know key information: The center is (-3,1) Since a2 = 4, a = 2, the major axis vertices are: (h + 2, k) and (h – 2, k) (-1, 1) and ( -5, 1) Since b2 = 1, b = 1, the minor axis vertices are: (h , k + 1) and (h , k + 1) (-3, 2) and ( -3, 0) 4. The foci can be found but are not needed to graph:c2 = a2 – b2 (2)2 – (1)2 = 3

  22. (-3, 2) (-5, 1) (-3, 0) (-1, 1) The center is (-3,1) The major axis vertices are: (-1, 1) and ( -5, 1) The minor axis vertices are:(-3, 2) and ( -3, 0) (-3, 1)

  23. Example: Find the center, vertices and foci of the ellipse: 4x2 + y2 + 8x +4y – 8 = 0 • Put the equation into standard form by completing the square. 2. Since the denominator of the y2-term is larger, the major axisis vertical.

  24. Example: Find the center, vertices and foci of the ellipse: 4x2 + y2 + 8x +4y – 8 = 0 Vertical 3. To graph this we need to know key information: The center is (1,-2) Since a2 = 16, a = 4, the major axis vertices are: (h , k + 4) and (h , k – 4) (1, 2) and ( 1, -6) Since b2 = 4, b = 2, the minor axis vertices are: (h +2 , k) and (h – 2, k) (3, -2) and ( -1, -2) 4. The foci can be found but are not needed to graph:c2 = a2 – b2 (4)2 – (2)2 = 12

  25. Eccentricity The eccentricity “e” of an ellipse is given by the ratio: Eccentricity is used to measure the “ovalness” of an ellipse. 0 < e < 1 for every ellipse. Because the foci of an ellipse are located along the major axis between the vertices and the center, it follows that: 0 < c < a For an ellipse that is nearly circular, the foci are close to the center and the ration c/a is small. For an elongated ellipse the foci are close to the vertices and the ratio c/a is close to 1.

  26. Eccentricity with Circles and Parabolas Ellipses have an eccentricity between zero and one. Circles are really ellipses with an eccentricity of zero. The two foci coincide and create the center. Parabolas always have an eccentricity of one.

  27. Another Note • The area of a circle is: π• r2 • The area of an ellipse is: π•a•b

  28. Wrap up…

  29. Ellipses Horizontal orientation Center (h, k) Vertex (h ± a, k) (major) Vertex (h, k ± b) (minor) Focus (h ± c, k) Length of major axis 2a Length of minor axis 2b a2 – b2 = c2 e = c/a Vertical orientation Center (h, k) Vertex (h, k ± a) (major) Vertex (h ± b, k) (minor) Focus (h, k ± c) Length of major axis 2a Length of minor axis 2b a2 – b2 = c2 e = c/a

  30. Homework 41 Page 722 # 1-6 all (matching), 7, 11,15, 21, 39 – 47 odd

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