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Hyperbolas

Hyperbolas. Definitions. A hyperbola is the set of all point P such that the difference of the distances between P and two fixed points, called the foci , is a constant. . The line through the foci intersects the hyperbola at the two vertices . The transverse axis joins the verticies .

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Hyperbolas

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  1. Hyperbolas

  2. Definitions Ahyperbolais the set of all point P such that the difference of the distances between P and two fixed points, called the foci, is a constant. The line through the foci intersects the hyperbola at the two vertices. Thetransverse axisjoins the verticies. Theconjugate axisjoins the “minor” verticies. Its midpoint is the hyperbola’s center.

  3. Hyperbolas They use this shape to construct nuclear power plants.

  4. Similarities and Differences

  5. Hyperbola w/ horizontal transverse axis. ( 0, b) Vertex (- a, 0) Vertex(a, 0) Asymptotes: Focus(- c, 0) Focus( c, 0) Vertices: ( 0, - b) Foci: Transverse Axis Standard form:

  6. Hyperbola w/ vertical transverse axis focus(0, c) Vertex (0, a) Transverse Axis Asymptotes: ( - b, 0 ) ( b, 0 ) Vertex (0, - a) Vertices: focus(0, - c) Foci: Standard form:

  7. Standard Equation of any hyperbola For the asymptotes just remember so whatever number is under the y – goes on top x – goes on bottom Asymptotes Equation Vertices AXIS Horizontal Vertical The foci lie on the transverse axis, c units from the center, where c2 = a2 + b2

  8. Write each hyperbola in standard form Hint: Set the equation equal to 1 !!! 1.4x2 – 9y2 = 36 2.4x2 – 16y2 = 64 3.100y2 – 36x2 = 3600 4.9x2 – y2 = 9 5.64y2 – x2 = 64 6.16x2 – 4y2 = 64

  9. Graph the equation of a hyperbola Graph 25y2 - 4x2 = 100. Identify the vertices, foci, and asymptotes of the hyperbola. Step 1 Rewrite the equation in standard form Write the original equation Divide each side by 100 Simplify

  10. Graph the equation of a hyperbola – cont Step 2 Identify the vertices, foci, and asymptotes. Find the vertices Write the formula for vertices h = 0 k = 0 a = 2 Label the h, k, and a values (0, 0 + 2) and (0, 0 -2) Plug in values Vertices are at (0, 2) and (0, -2) Simplify Find the Foci Write the formula for foci REMEMBER c2 = a2 + b2 Label the h, k, and c values h = 0 k = 0 c = 5.4 c2 = 22 + 52 = = 5.4 (0, 0 + 5.4) and (0, 0 -5.4) Plug in values (0, 5.4) and (0, -5.4) Foci are at Simplify

  11. Graph the equation of a hyperbola – cont Step 3 Graph the hyperbola. Decide whether it is a horizontal or vertical hyperbola Vertical (0, 2) and (0, -2) Plot the vertices. (0, 5.4) and (0, - 5.4) Plot the foci. = Draw the asymptotes This means that you go up 2 and right 5 from the center and also down 2 and right 5. Draw the hyperbola passing through the vertices and approaching the asymptotes.

  12. Graph the equation of a hyperbola Graph 9x2 – 25y2 = 225. Identify the vertices, foci, and asymptotes of the hyperbola. Step 1 Rewrite the equation in standard form Write the original equation Divide each side by 225 Simplify

  13. Graph the equation of a hyperbola – cont Step 2 Identify the vertices, foci, and asymptotes. Find the vertices Write the formula for vertices h = 0 k = 0 a = 5 Label the h, k, and a values (0 + 5, 0) and (0 - 5, 0) Plug in values (5, 0) and (-5, 0) Simplify Vertices are at Find the Foci Write the formula for foci REMEMBER c2 = a2 + b2 Label the h, k, and c values h = 0 k = 0 c = 5.8 c2 = 52 + 32 = 5.8 = (0 + 5.8, 0) and (0 – 5.8, 0) Plug in values (5.8, 0) and (-5.8, 0) Foci are at Simplify

  14. Graph the equation of a hyperbola – cont Step 3 Graph the hyperbola. Decide whether it is a horizontal or vertical hyperbola horizontal (5, 0) and (-5, 0) Plot the vertices. (5.8, 0) and (5.8, 0) Plot the foci. = Draw the asymptotes This means that you go up 3 and right 5 from the center and also down 3 and right 5. Draw the hyperbola passing through the vertices and approaching the asymptotes.

  15. Try These… Graph the hyperbola. Identify the vertices, foci, and asymptotes of the hyperbola. 1. Vertices (8, 3) and (-4, 3) Foci (8.3, 3) and (-4.3, 3) Asymptotes y = 1/3 and -1/3 2. Vertices (-3, 1) and (-3, -5) Foci (-3, 4.7) and (-3, -8.7) Asymptotes y = 1/2 and -1/2

  16. Write an equation of a hyperbola Write an equation of the hyperbola with the foci (-4,0) and (4, 0) and vertices at (-3,0) and (3,0). Solution Graph the information given Decide whether the graph is horizontal or vertical Write the standard form of the hyperbola Identify the variables: h, k, a, b h = 0 k = 0 a2 = 9 b2 =7 REMEMBER: a2 + b2 = c2 32 + b2 = 42 b2 = 7 = Plug into standard form

  17. Write an equation of a hyperbola Write an equation of the hyperbola with the foci (-3,0) and (3, 0) and vertices at (-1,0) and (1,0). Solution Graph the information given Decide whether the graph is horizontal or vertical Write the standard form of the hyperbola Identify the variables: h, k, a, b h = 0 k = 0 a2 = 1 b2 = 8 REMEMBER: a2 + b2 = c2 12 + b2 = 32 b2 = 8 = Plug into standard form

  18. Write an equation of a hyperbola Write an equation of the hyperbola with the foci (0,10) and (0,-10) and vertices at (0,-6) and (0,6). Solution Graph the information given Decide whether the graph is horizontal or vertical Write the standard form of the hyperbola Identify the variables: h, k, a, b h = 0 k = 0 a2 = 36 b2 = 64 REMEMBER: a2 + b2 = c2 62 + b2 = 102 b2 = 64 = Plug into standard form

  19. Thought question!!! In the conic section: what is the effect of switching the parameters 36 and 64 so that the equation becomes? A. The ellipse has a y – radius of 6 and an x – radius of 8 B. The ellipse has a y –radius of 8 and an x – radius of 6 • The hyperbola opens in the x – direction and has asymptotes • with slope of y = 3/4 and y = -3/4 D. The hyperbola opens in the y – direction and has asymptotes with slope of y = 4/3 and y = -4/3

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