Parametric forms of conics

What a parametric form is

• Instead of one equation linking $x$ and $y$, a parametric form gives $x$ and $y$ separately in terms of a third variable, the parameter (often $t$ or $\theta$).
• Each value of the parameter produces one point $(x, y)$ on the curve.
• As the parameter runs through its range, the point traces out the conic.

Standard parametric forms

• Circle $x^2 + y^2 = r^2$:

$x = r\cos\theta, \qquad y = r\sin\theta$

• Ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$:

$x = a\cos\theta, \qquad y = b\sin\theta$

• Parabola $y^2 = 4ax$:

$x = at^2, \qquad y = 2at$

• Rectangular hyperbola $xy = c^2$:

$x = ct, \qquad y = \frac{c}{t}$

• $\theta$ — for the circle and ellipse, an angle parameter.
• $t$ — for the parabola and hyperbola, a real-number parameter.

Converting parametric to Cartesian

• Eliminate the parameter to get back a single equation in $x$ and $y$.
• For circle/ellipse, use the identity $\cos^2\theta + \sin^2\theta = 1$.
• For the parabola/hyperbola, make the parameter the subject of one equation and substitute into the other.

Why parametric forms help

• They give an easy way to plot points — just feed in values of the parameter.
• They make some problems (like finding a general point on the curve for a tangent proof) much cleaner.