Parametric Equations 

In traditional Cartesian equations, a relationship is expressed between two variables—typically x and y. However, in many real-world scenarios, both variables depend on a third variable, often time. This is where parametric equations come in. Instead of one equation, we use two or more to define the coordinates as functions of an independent parameter.

1.0Parametric Equations Definition

A parametric equation defines a group of quantities as functions of one or more independent variables, called parameters. In two dimensions, parametric equations typically take the form:

$x=f(t),y=g(t)$

Here, t is the parameter, and f, g are parametric functions describing how x and y change over time.

2.0What are Parametric Equations with Examples?

Example 1:

Let: $x=cos(t),y=sin(t),0\le t\le 2\pi$

This defines a unit circle. As t varies from 0 to 2π, the coordinates (x, y) trace a circular path.

Example 2:

Projectile motion in physics is described by:

$x(t)=v_{0}cos(\theta )t,y(t)=v_{0}sin(\theta )t-\frac{1}{2}g{t}^2$

This pair of parametric equations models the position of a projectile over time.

3.0Parametric Equations and Curves

When you graph the ordered pairs (x(t), y(t)), the resulting plot is called a parametric curve. These curves are useful for:

• Describing motion and direction over time
• Modeling loops, spirals, and other shapes difficult to describe using y = f(x)
• Representing multi-variable systems

4.0Parametric Equations Formula

Standard Parametric Form (2D): x = f(t), y = g(t)   

Standard Parametric Equation of a Circle: $x=rcos(t),y=rsin(t)$

Parametric Equation of a Line:

If a line passes through point ($x_0,y_0$) and has direction vector ⟨a, b⟩, then the parametric form is:

$x=x_0+at,y=y_0+bt$

This is extremely useful in 2D and 3D geometry for describing the trajectory of points, particles, or lines in space.

5.0What is the Purpose of a Parametric Equation?

The purpose of a parametric equation is to:

• Represent complex curves and motions that are hard or impossible to express with Cartesian equations
• Track position over time (especially in physics and engineering)
• Describe paths with direction, orientation, or constraints
• Model real-world systems involving multiple dependent variables

In short, parametric equations extend the power of mathematics to describe dynamic systems with elegance and precision.

6.0Eliminating the Parameter

Sometimes, it's useful to convert parametric equations back to Cartesian form by eliminating the parameter.

Example:

Given: x = 2t + 1, y = 3t - 4 

Solve for t from the first equation: $t=\frac{x-1}{2}$

Substitute into the second: $y=3(\frac{x-1}{2})-4=\frac{3x-3}{2}-4=\frac{3x-11}{2}$

This gives the Cartesian form: $y=\frac{3x-11}{2}$

7.0Solved Examples on Parametric Equations 

Example 1: Given the parametric equations: $x=t^2,y=2t$

Eliminate the parameter t to find the Cartesian equation.

Solution: 

From y = 2t, solve for t:

$t=\frac{y}{2}$ 

Substitute into : $x=t^2$

 $x=(\frac{y}{2}{)}^2=\frac{y^2}{4}$

Answer: $x=\frac{y^2}{4}$

Example 2: Find the parametric equations of a line passing through point (2, -1) and having direction vector ⟨3, 4⟩.

Solution:

The general form is:

 $x=x_0+at,y=y_0+bt$

Substitute:

x = 2 + 3t, y = –1 + 4t 

Answer: x = 2 + 3t, y = -1 + 4t

Example 3: Given $x=t^2+1,y+t^3$, find $\frac{dy}{dx}$.

Solution:

First, compute derivatives:

 $\frac{dx}{dt}=2t,\frac{dy}{dt}=3t^2$

Then use:

 $\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}=\frac{3t^2}{2t}=\frac{3t}{2}$

Answer: $\frac{dy}{dx}=\frac{3t}{2}$

Example 4: Given: $x=cos(t),y=sin(t)$ for 0 ≤ t ≤ 2π, identify the curve.

Solution:

Use the identity:

${\cos }^2(t)+{\sin }^2(t)=1\Rightarrow x^2+y^2=1$

Answer: The curve is a circle of radius 1 centered at the origin.

Example 5: Write the parametric equations for a circle of radius 4 centered at (2, -3).

Solution:

The general form:

$x=h+rcos(t),y=k+rsin(t)$

Substitute h = 2, k = –3, r = 4:

$x=2+4cos(t),y=-3+4sin(t)$

Answer: $x=2+4cos(t),y=-3+4sin(t)$

8.0Practice Questions on Parametric Equations

1. Find the Cartesian equation by eliminating the parameter: $x=t^2,y=2t$
2. Graph the parametric equations: $x=cos(t),y=sin(2t),0\le t\le 2\pi$
3. Write the parametric equation of a line through (1, 2) with direction vector ⟨3, −1⟩
4. What shape is traced by: $x=2cos(t),y=2sin(t)$ for $0\le t\le 2\pi$?
5. A particle moves along a curve given by $x=t^3,y=t^2$. Find $\frac{dy}{dx}$ in terms of t