Edexcel IAL - Further Pure Mathematics 1- 4.2 Parametric Equations of Parabolas and Hyperbolas- Study notes  - New syllabus

Idea of Parametric Equations for the Parabola and Rectangular Hyperbola

In many situations, it is convenient to describe a curve not directly in terms of \( x \) and \( y \), but in terms of a third variable called a parameter, usually denoted by \( t \). A parametric equation expresses both \( x \) and \( y \) in terms of \( t \).

This gives a simple way to generate every point on a curve.

Parametric Idea for the Parabola

Consider the parabola:

\( y^2 = 4ax \)

A very convenient way to describe all points on this parabola is to use the parameter \( t \) and write:

\( x = at^2,\quad y = 2at \)

So the general point on the parabola is:

\( (at^2,\ 2at) \)

As \( t \) takes different values, this point moves along the parabola.

For example:

• If \( t = 0 \), the point is \( (0,0) \).
• If \( t = 1 \), the point is \( (a, 2a) \).
• If \( t = -1 \), the point is \( (a, -2a) \).

Thus, varying \( t \) traces out the whole parabola.

Why This Works

Substituting \( x = at^2 \) and \( y = 2at \) into \( y^2 = 4ax \) gives:

\( y^2 = (2at)^2 = 4a^2t^2 \)

\( 4ax = 4a(at^2) = 4a^2t^2 \)

So the parametric point always satisfies the equation of the parabola.

Parametric Idea for the Rectangular Hyperbola

The rectangular hyperbola:

\( xy = c^2 \)

can be written in parametric form as:

\( x = ct,\quad y = \dfrac{c}{t} \)

So the general point is:

\( (ct,\ \dfrac{c}{t}) \)

As \( t \) varies (except 0), the point moves along the hyperbola.

Why This Works

Substituting gives:

\( xy = ct \cdot \dfrac{c}{t} = c^2 \)

So the parametric form always satisfies the Cartesian equation.

Geometric Interpretation

• The parameter \( t \) controls the position of the point on the curve.
• Changing \( t \) moves the point smoothly along the curve.
• Parametric equations give a systematic way to describe all points on a curve.