FAD1014 L32 — Parametric Equations

Parametric equations express $x$ and $y$ coordinates as functions of a third independent variable (parameter $t$ or $\theta$). This lecture covers converting parametric equations to Cartesian form for conic sections.

Learning Outcomes

  1. Understand that $x$ and $y$ can be expressed as functions of a third independent variable
  2. Convert parametric equations to Cartesian equation
  3. Identify and describe the locus type with its details
  4. Sketch the locus obtained

Parametric Representation: Circle

Standard parametric equations: $$x = r \cos t \quad \text{and} \quad y = r \sin t$$

Cartesian form derivation: $$\cos t = \frac{x}{r}, \quad \sin t = \frac{y}{r}$$

Using $\cos^2 t + \sin^2 t = 1$: $$\frac{x^2}{r^2} + \frac{y^2}{r^2} = 1 \implies x^2 + y^2 = r^2$$

General form (center at $(h, k)$): $$x - h = r \cos t \quad \text{and} \quad y - k = r \sin t$$

Converts to: $$(x - h)^2 + (y - k)^2 = r^2$$

Parametric Representation: Parabola

Horizontal Parabola (opens right/left)

Parametric equations: $$x - h = at^2 \quad \text{and} \quad y - k = 2at$$

Cartesian form derivation: From $y - k = 2at \implies t = \frac{y - k}{2a}$

Substituting: $$x - h = a\left(\frac{y - k}{2a}\right)^2 = \frac{(y - k)^2}{4a}$$

$$(y - k)^2 = 4a(x - h)$$

Vertical Parabola (opens up/down)

Parametric equations: $$x - h = 2at \quad \text{and} \quad y - k = at^2$$

Cartesian form: $$(x - h)^2 = 4a(y - k)$$

Parametric Representation: Ellipse

Parametric equations: $$x - h = a \cos t \quad \text{and} \quad y - k = b \sin t$$

Cartesian form derivation: $$\cos t = \frac{x - h}{a}, \quad \sin t = \frac{y - k}{b}$$

Using $\cos^2 t + \sin^2 t = 1$: $$\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$$

Parametric Representation: Hyperbola

Vertical Hyperbola (opens up/down)

Parametric equations: $$x - h = a \sec t \quad \text{and} \quad y - k = b \tan t$$

Cartesian form derivation: Using $1 + \tan^2 t = \sec^2 t$: $$\frac{(y - k)^2}{b^2} - \frac{(x - h)^2}{a^2} = 1$$

Horizontal Hyperbola (opens left/right)

Parametric equations: $$x - h = a \tan t \quad \text{and} \quad y - k = b \sec t$$

Cartesian form: $$\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1$$

Practice Problems

Problem 1

A curve has parametric equations $x = \sin t - 2$ and $y = \cos t + 1$, where $t \in \mathbb{R}$.

Show that the Cartesian equation is a circle and sketch the graph.

Problem 2

A curve has parametric equations $x = \cot t + 1$ and $y = \csc^2 t - 4$, where $0 < t < \pi$.

Show that the Cartesian equation is a parabola, sketch the graph, and state the domain and range.

Problem 3

Find the standard form of the Cartesian equation:

(a) $x = 2t + 3$, $y = t^2$

(b) $x = 2\cos\theta$, $y = 3\sin\theta$

(c) $x = 5\sin\theta$, $y = 4\cos\theta$

(d) $x = \sec^2 t - 1$, $y = \tan t$

(e) $x = 2\sec t$, $y = \tan t - 8$

Key Identities Used

Identity Application
$\cos^2 t + \sin^2 t = 1$ Circle, Ellipse
$1 + \tan^2 t = \sec^2 t$ Hyperbola
$1 + \cot^2 t = \csc^2 t$ Parabola

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