# 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 yk:

(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 equation of an ellipse $$\dfrac{(x-h)^2}{a^2}+\dfrac{(y-k)^2}{b^2}=1$$ centered at (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 xh 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,

The equation of a hyperbola $$\dfrac{(x-h)^2}{a^2}-\dfrac{(y-k)^2}{b^2}=1$$ centered at (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:

$x = a\tan\theta + h$

$y = b\sec\theta + k$