# 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,

*h*,

*k*) in parametric form is:

\[x = a\cos\theta + h\]

\[y = b\cos\theta + k\]

If *a* = *b* = *r*, the parametric equation reduces to an equation of a circle of radius *r*.

## 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,

*h*,

*k*) in parametric form is:

\[x = a\sec\theta + h\]

\[y = b\tan\theta + k\]

If the hyperbola is defined by the equation \(-\dfrac{(x-h)^2}{a^2}+\dfrac{(y-k)^2}{b^2}=1\), the equation of the hyperbola with a vertical transverse axis parallel to the *y*-axis in parametric form is:

\[x = a\tan\theta + h\]

\[y = b\sec\theta + k\]