Eliminating the Parameter — Definition, Formula & Examples

Eliminating the parameter is the process of removing the parameter (usually

t

) from a pair of parametric equations

x(t)

and

y(t)

to produce a single equation in

x

and

y

alone.

Given parametric equations $x = f(t)$ and $y = g(t)$ , eliminating the parameter means solving one equation for $t$ (or using an identity to relate $f$ and $g$ ) and substituting into the other to obtain an equivalent rectangular equation $F(x, y) = 0$ .

How It Works

There are two main strategies. If one parametric equation can be solved for

t

algebraically, substitute that expression into the other equation. When trigonometric parametric equations involve

\sin t

and

\cos t

, use the Pythagorean identity

\sin^2 t + \cos^2 t = 1

to eliminate

t

without solving for it directly. After elimination, you may need to note any domain restrictions that the original parameter imposed on

x

y

Worked Example

Problem: Eliminate the parameter from the parametric equations x = 2cos(t) and y = 3sin(t).

Step 1: Isolate the trig functions by dividing each equation:

\cos t = \frac{x}{2}, \quad \sin t = \frac{y}{3}

Step 2: Apply the Pythagorean identity

\cos^2 t + \sin^2 t = 1

\left(\frac{x}{2}\right)^2 + \left(\frac{y}{3}\right)^2 = 1

Step 3: Simplify to standard form:

\frac{x^2}{4} + \frac{y^2}{9} = 1

Answer: The rectangular equation is

\dfrac{x^2}{4} + \dfrac{y^2}{9} = 1

, an ellipse centered at the origin.

Why It Matters

Converting parametric curves to rectangular form lets you identify familiar shapes (circles, ellipses, parabolas) and graph them without a table of

t

-values. This technique appears frequently in precalculus, AP Calculus BC, and physics courses whenever motion is described parametrically.

Common Mistakes

Mistake: Forgetting domain restrictions after eliminating the parameter.

Correction: The original parameter may restrict

x

y

to a subset of the rectangular curve. For example, if

t \geq 0

and

x = t

, then

x \geq 0

, so only part of the rectangular curve applies.

Related Terms

Area of a Parametric Curve — Uses parametric form to compute enclosed area
Equation of a Circle — Common result after eliminating trig parameters
Cauchy-Riemann Equations — Advanced use of parametric and complex variables
Bernoulli Differential Equation — Substitution technique parallels parameter elimination