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Conics: Hyperbolas: Finding Information
     From the Equation
(page 2 of 3)

Sections: Introduction, Finding information from the equation, Finding the equation from information


  • Find the center, vertices, foci, eccentricity, and asymptotes of the hyperbola with the given equation, and sketch:  (y^2) / 25 - (x^2) / 144 = 1
  • Since the y part of the equation is added, then the center, foci, and vertices will be above and below the center (on a line paralleling the y-axis), rather than side by side.

    Looking at the denominators, I see that a2 = 25 and b2 = 144, so a = 5 and b = 12. The equation c2 a2 = b2 tells me that c2 = 144 + 25 = 169, so c = 13, and the eccentricity is e = 13/5. Since x2 = (x 0)2 and y2 = (y 0)2, then the center is at (h, k) = (0, 0). The vertices and foci are above and below the center, so the foci are at (0, 13) and (0, 13), and the vertices are at (0, 5) and (0, 5).

     
    Because the
    y part of the equation is dominant (being added, not subtracted), then the slope of the asymptotes has the a on top, so the slopes will be m = 5/12. To graph, I start with the center, and draw the asymptotes through it, using dashed lines:

     

     

    axis system, from -50 to 50, with center and asymptotes marked 

     

     
    Then I draw in the vertices, and rough in the graph, rotating the paper as necessary and "eye-balling" for smoothness:

     

    vertices added, and branches roughed in

     

     

    Then I draw in the final graph as a neat, smooth, heavier line:

     

    final graph, with 'answer' line drawn bold and clear, but neatly

    And the rest of my answer is:

      center (0, 0), vertices (0, 5) and (0, 5), foci (0, 13) and (0, 13),
      eccentricity e = 13/5, and asymptotes y = +/- (5/12) x

  • Give the center, vertices, foci, and asymptotes for the hyperbola
    with equation:   [(x + 3)^2] / 16 - [(y - 2)^2] / 9 = 1
  • Since the x part is added, then a2 = 16 and b2 = 9, so a = 4 and b = 3. Also, this hyperbola's foci and vertices are to the left and right of the center, on a horizontal line paralleling the x-axis.

    From the equation, clearly the center is at (h, k) = (3, 2). Since the vertices are a = 4 units to either side, then they are at (7, 2) and at (1, 2). The equation c2 a2 = b2 gives me c2 = 9 + 16 = 25, so c = 5, and the foci, being 5 units to either side of the center, must be at (8, 2) and (2, 2). Copyright Elizabeth Stapel 2010-2011 All Rights Reserved

    Since the a2 went with the x part of the equation, then a is in the denominator of the slopes of the asymptotes, giving me m = 3/4. Keeping in mind that the asymptotes go through the center of the hyperbola, the asymptes are then given by the straight-line equations  y 2 =  (3/4)(x + 3).

      center (3, 2), vertices (7, 2) and (1, 2), foci (8, 2) and (2, 2),
      and asymptotes y = +/- (3/4) (x + 3) + 2

  • Find the center, vertices, and asymptotes of the hyperbola with equation
    4x2 5y2 + 40x 30y 45 = 0.
  • To find the information I need, I'll first have to convert this equation to "conics" form by completing the square.

      4x2 + 40x 5y2 30y = 45
      4(x2 + 10x        ) 5(y2 + 6y       ) = 45 + 4(   ) 5(  )

      4(x2 + 10x + 25) 5(y2 + 6y + 9) = 45 + 4(25) 5(9)

      4(x + 5)2 5(y + 3)2 = 45 + 100 45

      [4(x + 5)^2] / 100 - [5(y + 3)^2] / 100 = 100 / 100

      [(x + 5)^2] / 25 - [(y +3)^2] / 20 = 1

    Then the center is at (h, k) = (5, 3). Since the x part of the equation is added, then the center, foci, and vertices lie on a horizontal line paralleling the x-axis; a2 = 25 and b2 = 20, so a = 5 and b = 2sqrt[5]. The equation a2 + b2 = c2 gives me c2 = 25 + 20 = 45, so c = sqrt[45] = 3sqrt[5]. The slopes of the two asymptotes will be m = (2/5)sqrt[5]. Then my complete answer is:

      center (5, 3), vertices (10, 3) and (0, 3),

      foci (-5 - 3sqrt[5], -3) and (-5 + 3sqrt[5], -3),

      and asymptotes y = +/- (2sqrt[5] / 5) (x + 5) - 3

If I had needed to graph this hyperbola, I'd have used a decimal approximation of 0.89442719... for the slope, but would have rounded the value to something reasonable like m =  0.9.

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Cite this article as:

Stapel, Elizabeth. "Conics: Hyperbolas: Finding Information From the Equation." Purplemath.
    Available from http://www.purplemath.com/modules/hyperbola2.htm.
    Accessed
 

 



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