Activities

Conic sections - ellipse and hyperbola - cartesian equation in standard form

The standard forms of the equations for the ellipse and hyperbola assume that:

Study the diagrams 3.36 p109, 3.42 p113, 3.53 p121 and 3.55 p123 QMaths12C.

The symmetry of the ellipse and hyperbola means that there are two possible foci and two possible directrices.

When the foci are on the x-axis, they are (±ae,0). The directrices are x = ±a/e. You may wish to skip the demonstration of these results on p108 QMaths12C.

A note is provided showing the derivation of the standard forms.

Given the equation, you need to be able to find the:

  • centre
  • vertices
  • foci
  • equations of directrices
  • lengths of major and minor axes (ellipse)
  • lengths of transverse and conjugate axes (hyperbola)
  • equations of asymptotes (hyperbola).
    • When the centre of the ellipse or hyperbola is not at the origin, the standard form can be adjusted to allow for translations (example).

    Attempt problems Ex 3.3 p115 and Ex 3.4 (3cgh tricky) p126 QMaths12C.

    To explore asymptotes of the hyperbola, use the Famous Curves Applet Index. What happens if you:

    Explain the results.

    The ellipse and hyperbola are not functions (why?). To plot the curve using "y = ...", it is necessary to rearrange the equation as two separate functions. Choose some equations and try this.