• Algebra > Parametric Equations of Conics

PARAMETRIC EQUATIONS OF CONICS

The conic sections can be represented by parametric equations. This page shows how one derives the parametric equations of the conic sections.

The Circle and Ellipse

The equation of an ellipse centered at (h, k) in standard form is: \(\dfrac{(x-h)^2}{a^2}+\dfrac{(y-k)^2}{b^2}=1\). To express in parametric form, begin by solving for y – k:

(i) \(y-k = \frac{b}{a}\sqrt{a^2-(x-h)^2}\)

Now, let \(x - h = a\cos\theta\). Then,

(ii) \(y-k = \frac{b}{a}\sqrt{a^2 - a^2\cos^{2}\theta} = b\sin\theta\)

Therefore,

The Hyperbola

The equation of a hyperbola centered at (h, k) in standard form is: \(\dfrac{(x-h)^2}{a^2}-\dfrac{(y-k)^2}{b^2}=1\) or \(-\dfrac{(x-h)^2}{a^2}+\dfrac{(y-k)^2}{b^2}=1\). To express in parametric form, begin by solving for x – h in the first equation:

(i) \(x-h = \frac{a}{b}\sqrt{b^2+(y-k)^2}\)

Now, let \(y - k = b\tan\theta\). Then,

(ii) \(x-h = \frac{a}{b}\sqrt{b^2 + b^2\tan^{2}\theta} = a\sec\theta\)

Therefore,