Conic Sections: Ellipse, Parabola and Hyperbola · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Slicing a cone into three curves

Mathematics · Geometry · ages 17-18
Name ______________________   Date ____________

  1. A parabola has focus F and directrix line d. Which points lie on it?

    • Points the same distance from F as from d
    • Points twice as far from F as from d
    • Points whose distances from F add to a fixed number
    • Points whose distances from two fixed points differ by a fixed amount

  2. The curve x^2/25 + y^2/9 = 1 is an ellipse. What are its vertices?

    • (5, 0) and (-5, 0)
    • (0, 5) and (0, -5)
    • (3, 0) and (-3, 0)
    • (4, 0) and (-4, 0)

  3. The curve x squared over 25 plus y squared over 9 equals 1 is an ellipse. What are its vertices?

    • (5, 0) and (-5, 0)
    • (0, 5) and (0, -5)
    • (3, 0) and (-3, 0)

  4. An ellipse has foci F1 and F2. Which property defines it?

    • The difference of the distances stays fixed
    • The distance to F1 equals the distance to a fixed line
    • The sum of the distances to F1 and F2 stays fixed

  5. The parabola y squared equals 12x has its vertex at the origin. How far is its focus from the vertex?

    • 12
    • 6
    • 3

  6. The parabola y^2 = 12x has its vertex at the origin. How far is its focus from the vertex?

    Answer: ______________

  7. The hyperbola x^2/16 - y^2/9 = 1 has foci at distance c from the centre. What is c?

    Answer: ______________

  8. The hyperbola x squared over 16 minus y squared over 9 equals 1 has foci at distance c from the centre. What is c?

    Answer: ______________

  9. Which equation names a hyperbola opening left and right?

    • x squared over 16 minus y squared over 9 equals 1
    • x squared over 16 plus y squared over 9 equals 1
    • y squared equals 12x

  10. After completing the square, a curve becomes (x plus 2) squared over 9 plus (y minus 3) squared over 4 equals 1. Give the x-coordinate of its centre.

    Answer: ______________

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Answer key

For grown-ups. Fold this page away before handing over the rest.

Slicing a cone into three curves W1-mt_V4YSI3a_Eu-s1

  1. Points the same distance from F as from d · Equal distance to focus and directrix is the defining property of a parabola.
  2. (5, 0) and (-5, 0) · The bigger denominator 25 sits under x, so a = 5 along the x-axis.
  3. (5, 0) and (-5, 0) · The bigger denominator 25 sits under x, so a equals 5 along the x-axis.
  4. The sum of the distances to F1 and F2 stays fixed · A fixed distance sum is the ellipse rule, and the difference rule gives a hyperbola.
  5. 3 · Matching y squared equals 4px gives 4p equals 12, so p equals 3.
  6. 3 · 4p = 12 gives p = 3, the focus to vertex distance.
  7. 5 · c squared = 16 + 9 = 25, so c = 5.
  8. 5 · c squared equals 16 plus 9, which is 25, so c equals 5.
  9. x squared over 16 minus y squared over 9 equals 1 · Only the minus between the squares splits the curve into two branches.
  10. -2 · The form (x minus h) squared centres at h, and x plus 2 means h equals minus 2.
Worksheet · LightMySky