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Mathematics Dictionary
Dr. K. G. Shih
Conic Section : Ellipse
Questions


  • Q01 | - Quiz about ellipse
  • Q02 | - Answer to Quiz
    • Answers


    Q01. Quiz in ellipse

    1. Find the locus of the following equations :

    • a. (x-2)^2/5^2 + (y+3)^2/3^2 = 1.
    • b. 9*x^2 + 25*y^2 + 54*x + 100*y + 181 = 0
    • c. 9*x^2 + 25*y^2 + 54*x + 100*y - 44 = 0
    • d. 9*x^2 + 25*y^2 + 54*x + 100*y + 200 = 0
    • e. 9*x^2 + 20*x*y + 25*y^2 + 18*x + 100*y - 200 = 0
    2. Compare (x-2)^2/5^2 + (y+3)^2/3^2 = 1 with (x-2)^2/3^2 + (y+3)^2/4^2 = 1.

    • a. Principal axis.
    • b. Focal length f.
    • c. Coordinate of Foci.
    • d. Vertice on principal axis
    • e. Polar form
    • f. Equation of directrix = 0
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    Q02. Answer to Quiz in ellipse

    1. Find the locus of the following equations :
    • a. (x-2)^2/5^2 + (y+3)^2/3^2 = 1.
      • It is an ellipse.
      • Principal axis y = -3.
    • b. 9*x^2 + 25*y^2 + 54*x + 100*y + 181 = 0
      • 9*(x+3)^2 - 81 + 25*(y+4)^2 -100 + 181 = 0.
      • 9*(x+3)^2 + 25*(y+2)^2 = 0.
      • It is a point.
      • Principal axis y = -3.
    • c. 9*x^2 + 25*y^2 + 54*x + 100*y - 116 = 0
      • 9*(x+3)^2 - 81 + 25*(y+4)^2 - 100 - 44 = 0.
      • 9*(x+3)^2 + 25*(y+2)^2 = 225.
      • (x+3)^2/25 + (y+2)^2/9 = 1
      • It is an ellipse.
      • Principal axis y = -2.
    • d. 9*x^2 + 25*y^2 + 18*x + 100*y + 200 = 0
      • 9*(x+3)^2 - 81 + 25*(y+4)^2 - 100 + 200 = 0.
      • 9*(x+3)^2 + 25*(y+2)^2 = -19.
      • No locus in real number system.
    • e. 9*x^2 + 20*x*y + 25*y^2 + 18*x + 100*y - 200 = 0
      • Since B^2 - 4*A*C = 20^2 - 4*9*25 = -500
      • Hence it is an ellipse.
      • Or a point.
      • Or no locus in real system.
    2. Compare (x-2)^2/5^2 + (y+3)^2/3^2 = 1 with (x-2)^2/3^2 + (y+3)^2/4^2 = 1.
    • a. Principal axis.
    • b. Focal length f.
    • c. Coordinate of Foci.
    • d. Vertice on principal axis
    • e. Polar form
    • f. Equation of directrix = 0
    For (x-2)^2/5^2 + (y+3)^2/3^2 = 1.
    • Principal axis is y = - 3.
    • Focal length f = Sqr(a^2-b^2) = 4.
    • Center is at (2,-3).
    • Coordinate of foci are (-2,-3) and (6,-3).
    • Coordinate of vertice are (-3,-3) and (7,-3).
    • Polar form : R = D*e/(1-e*cos(A)) or R = D*e/(1+e*cos(A)).
    • Equation of directrix
      • Since e = f/a = 4/5 = 0.8.
      • R = a - f = D*e/(1+e) when A = 180 degrees.
      • Hence D*e = (a-f)*(1+e) = (5-4)*(1+0.8) = 1.8
      • Hence D = 1.8/e = 2.25.
      • Equation of directrix at x = -3 - 2.25 = - 5.25 for focus (-3,-3).
      • Equation of directrix at x = 7 + 2.25 = 9.25 for focus (7,-3).
    For (x-2)^2/3^2 + (y+3)^2/5^2 = 1.
    • Principal axis is x = 2.
    • Focal length f = Sqr(a^2-b^2) = 4.
    • Center is at (2,-3).
    • Coordinate of foci are (2,-7) and (2,1).
    • Coordinate of vertice are (2,-8) and (2,2).
    • Polar form : R = D*e/(1-e*sin(A)) or R = D*e/(1+e*sin(A)).
    • Equation of directrix
      • Since e = f/a = 4/5 = 0.8.
      • R = a - f = D*e/(1+e) when A = 270 degrees.
      • Hence D*e = (a-f)*(1+e) = (5-4)*(1+0.8) = 1.8
      • Hence D = 1.8/e = 2.25.
      • Equation of directrix at y = -7 - 2.25 = - 9.25 for focus (2,-7).
      • Equation of directrix at x = 1 + 2.25 = 3.25 for focus (2,1).
    Go to Begin
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