| Content On This Page | ||
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| Parametric Representation and its Utility | Parametric Equations for Circle, Parabola, Ellipse, and Hyperbola | |
Parametric Equations of Conics (Consolidated)
Parametric Representation and its Utility
In coordinate geometry, curves are typically described by a single equation relating the $x$ and $y$ coordinates, known as the Cartesian equation (e.g., $y=f(x)$, $x=g(y)$, or $F(x,y)=0$). However, there is an alternative way to represent a curve using parametric equations. This approach introduces a third variable, called a parameter, and expresses both $x$ and $y$ coordinates as functions of this parameter.
What are Parametric Equations?
Instead of defining a curve by an equation in $x$ and $y$, parametric equations define the coordinates of each point on the curve using a single, independent variable. Let this parameter be denoted by $t$ (though any variable can be used, $\theta$ is common for circular or elliptical forms, $t$ for general curves or time).
The parametric representation of a curve is given by a set of equations, usually one for each coordinate:
$\mathbf{x = f(t)}$
$\mathbf{y = g(t)}$
where $f$ and $g$ are functions of the parameter $t$. The parameter $t$ varies over a specified range or interval of real numbers. As $t$ takes on values within this interval, the pair of equations $(x(t), y(t))$ generates the coordinates of points that together form the curve.
For example, the parametric equations $x = \cos t$, $y = \sin t$ for $0 \le t < 2\pi$ represent a circle centered at the origin with radius 1. Each value of $t$ corresponds to a unique point on the circle.
Utility of Parametric Equations
Parametric representations are not just an alternative way of writing equations; they offer significant advantages and are particularly useful in various contexts, especially in mathematics, physics, and computer graphics.
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Describing Motion and Paths: Parametric equations are ideal for describing the path of a moving object. If the parameter $t$ represents time, then $x(t)$ and $y(t)$ give the horizontal and vertical positions of the object at any given time $t$. This allows us to analyze the trajectory, velocity, and acceleration of the object along the path.
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Representing Non-Function Curves: Many important curves, such as circles, ellipses, and hyperbolas, cannot be represented by a single function of the form $y=f(x)$ because they fail the vertical line test (i.e., for a single x-value, there might be multiple y-values). Parametric equations overcome this limitation by expressing $x$ and $y$ independently in terms of a parameter.
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Ease of Generating Points: To plot a curve from its parametric equations, one can simply choose various values for the parameter $t$ within its range, calculate the corresponding $x$ and $y$ values, and plot the points $(x, y)$. This is often much simpler than solving a complex Cartesian equation for $y$ in terms of $x$ (or vice-versa).
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Direction and Orientation: Parametric equations implicitly define a direction or orientation along the curve. As the parameter $t$ increases, the point $(x(t), y(t))$ moves along the curve in a specific direction. This is particularly important when dealing with trajectories or oriented curves.
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Simplification in Calculus: Calculating slopes of tangent lines ($\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$), arc lengths ($\int \sqrt{(dx/dt)^2 + (dy/dt)^2} dt$), and other calculus properties is often much easier using parametric derivatives and integrals compared to working directly with a complex Cartesian equation or implicit differentiation.
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Standard Forms for Conics: As we will see, standard parametric forms exist for circles, parabolas, ellipses, and hyperbolas. These forms utilize trigonometric or hyperbolic identities and provide a consistent way to represent points on these curves, which can simplify certain geometric problems or proofs.
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Relationship to Vectors: Parametric equations are closely related to vector representation of curves, where the position vector of a point on the curve is given as a function of the parameter: $\vec{r}(t) = x(t)\hat{i} + y(t)\hat{j}$.
In summary, parametric representation is a versatile and powerful tool in geometry and calculus, offering advantages in terms of flexibility, computation, and the ability to model dynamic situations.
Parametric Representation (Summary)
Definition:
Representing coordinates $(x, y)$ of a curve as functions of a single parameter $t$ (or $\theta$).
$\mathbf{x = f(t), \quad y = g(t)}$
Parameter:
An independent variable, typically $t$ (time, or general) or $\theta$ (angle).
Utility:
- Describes motion/trajectories.
- Represents non-function curves easily.
- Facilitates plotting.
- Defines curve orientation.
- Simplifies calculus calculations (slope, length).
- Provides standard forms for conic sections.
Conversion to Cartesian:
Eliminate the parameter $t$ from the two equations to get an equation involving only $x$ and $y$.
Parametric Equations for Circle, Parabola, Ellipse, and Hyperbola
For each of the non-degenerate conic sections—circle, parabola, ellipse, and hyperbola—standard parametric representations exist. These equations express the $x$ and $y$ coordinates of any point on the conic as functions of a single parameter, leveraging trigonometric or hyperbolic identities that match the structure of their Cartesian equations.
Summary Table of Standard Parametric Equations
The following table provides standard parametric equations for conics centered at the origin or with vertex at the origin, unless otherwise specified. The parameters are typically denoted by $\theta$ for trigonometric forms and $t$ for other forms. Note that for ellipses and hyperbolas, $a$ and $b$ refer to the semi-axis lengths as defined in their standard Cartesian equations.
| Conic Section | Standard Cartesian Equation | Standard Parametric Equations | Parameter | Typical Range / Notes |
|---|---|---|---|---|
| Circle | $(x-h)^2 + (y-k)^2 = r^2$ | $x = h + r \cos \theta$ $y = k + r \sin \theta$ |
$\theta$ (Angle) | $0 \le \theta < 2\pi$ to trace the entire circle. |
| $x^2 + y^2 = r^2$ (Center at Origin) | $x = r \cos \theta$ $y = r \sin \theta$ |
$\theta$ (Angle) | $0 \le \theta < 2\pi$ | |
| Parabola | $y^2 = 4ax$ (Vertex at Origin, $a>0$) | $x = at^2$ $y = 2at$ |
$t$ | $t \in \mathbb{R}$ (any real number). Traces the entire parabola. |
| $x^2 = 4ay$ (Vertex at Origin, $a>0$) | $x = 2at$ $y = at^2$ |
$t$ | $t \in \mathbb{R}$. | |
| $(y-k)^2 = 4a(x-h)$ (Vertex $(h,k)$, $a>0$) | $x = h + at^2$ $y = k + 2at$ |
$t$ | $t \in \mathbb{R}$. | |
| Ellipse | $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ (Center at Origin) | $x = a \cos \theta$ $y = b \sin \theta$ |
$\theta$ (Eccentric Angle) | $0 \le \theta < 2\pi$ to trace the entire ellipse. |
| $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$ (Center $(h,k)$) | $x = h + a \cos \theta$ $y = k + b \sin \theta$ |
$\theta$ (Eccentric Angle) | $0 \le \theta < 2\pi$ | |
| Hyperbola | $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ (Center at Origin) | $x = a \sec \theta$ $y = b \tan \theta$ |
$\theta$ (Angle) | $0 \le \theta < 2\pi$, $\theta \neq \frac{\pi}{2}, \frac{3\pi}{2}$. Traces both branches. |
| $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$ (Center at Origin) | $x = b \tan \theta$ $y = a \sec \theta$ |
$\theta$ (Angle) | $0 \le \theta < 2\pi$, $\theta \neq \frac{\pi}{2}, \frac{3\pi}{2}$. | |
| Hyperbola (Alternative - Hyperbolic Functions) | $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ (Center at Origin) | $x = a \cosh t$ $y = b \sinh t$ |
$t$ (Hyperbolic parameter) | $t \in \mathbb{R}$. Traces the right branch ($x \ge a$). For the left branch ($x \le -a$), use $x = -a \cosh t$. |
Example 1. Write the parametric equations for the ellipse $4x^2 + 9y^2 = 36$.
Answer:
The given equation is $4x^2 + 9y^2 = 36$.
To find the parametric equations, we first convert the equation to the standard form $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$. This involves making the right side of the equation equal to 1 by dividing all terms by 36:
$\frac{4x^2}{36} + \frac{9y^2}{36} = \frac{36}{36}$
... (i)
Simplify the fractions in equation (i):
$\frac{x^2}{9} + \frac{y^2}{4} = 1$
... (ii)
Equation (ii) is in the standard form of an ellipse centered at the origin $(0, 0)$. Comparing it with $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, we identify the denominators under $x^2$ and $y^2$ as $a^2$ and $b^2$ (where $a^2$ is the larger denominator, 9, and $b^2$ is the smaller, 4, for the standard ellipse where $a>b$; however, the parametric form $x=A \cos \theta, y=B \sin \theta$ applies to $\frac{x^2}{A^2} + \frac{y^2}{B^2} = 1$ where $A, B$ are just the semi-axes lengths along x and y respectively).
Here, $\frac{x^2}{3^2} + \frac{y^2}{2^2} = 1$.
So, the semi-axis length along the x-direction is 3, and the semi-axis length along the y-direction is 2.
Using the standard parametric form $x = (\text{semi-axis length along x}) \cos \theta$ and $y = (\text{semi-axis length along y}) \sin \theta$:
$\mathbf{x = 3 \cos \theta}$
... (iii)
$\mathbf{y = 2 \sin \theta}$
... (iv)
Equations (iii) and (iv) are the parametric equations for the ellipse $4x^2 + 9y^2 = 36$, where $\theta$ is the parameter, typically varying from $0$ to $2\pi$.
Example 2. Identify the conic represented by the parametric equations $x = t^2$, $y = 4t$.
Answer:
Given the parametric equations:
$x = t^2$
... (1)
$y = 4t$
... (2)
To identify the conic, we need to find the Cartesian equation by eliminating the parameter $t$.
From equation (2), it is easy to express $t$ in terms of $y$:
$t = \frac{y}{4}$
... (iii)
Now, substitute the expression for $t$ from equation (iii) into equation (1):
$x = \left(\frac{y}{4}\right)^2$
... (iv)
Simplify equation (iv):
$x = \frac{y^2}{16}$
... (v)
Rearrange equation (v) to match a standard form:
$y^2 = 16x$
... (vi)
Equation (vi) is the Cartesian equation. This is the standard equation of a parabola of the form $y^2 = 4ax$, with $4a = 16$, so $a = 4$. This parabola has its vertex at the origin and opens to the right.
The conic represented by the parametric equations $x = t^2$, $y = 4t$ is a $\mathbf{parabola}$.
Note: While the standard parametric form for $y^2 = 16x$ is $x = 4T^2, y = 8T$ (using a new parameter $T$), the given equations $x=t^2, y=4t$ represent the same curve. For $y^2 = 16x$, a point $(x, y)$ satisfies the equation. The point $(t^2, 4t)$ satisfies $(4t)^2 = 16t^2$, $16t^2 = 16(t^2)$, which is true. So, $(t^2, 4t)$ is on the parabola $y^2 = 16x$. Different parameterizations of the same curve are possible.
Parametric Equations of Conic Sections (Summary)
General Approach:
Find functions $x(t), y(t)$ (or $x(\theta), y(\theta)$) such that substituting them into the Cartesian equation results in an identity.
Key Identities Used:
- $\cos^2 \theta + \sin^2 \theta = 1$ (Circle, Ellipse)
- $\sec^2 \theta - \tan^2 \theta = 1$ (Hyperbola)
- $\cosh^2 t - \sinh^2 t = 1$ (Hyperbola)
- $y^2 = 4ax$ or $x^2 = 4ay$ structure (Parabola)
Standard Forms (Vertex/Center at Origin):
See the table provided earlier for common parametric equations for standard orientations.
For General Conics (Center $(h,k)$ or Vertex $(h,k)$):
Add $h$ to the x-equation and $k$ to the y-equation of the corresponding origin-based parametric form. E.g., $(x-h)^2 + (y-k)^2 = r^2 \implies x=h+r \cos\theta, y=k+r \sin\theta$.
Converting to Cartesian:
Eliminate the parameter by solving one equation for the parameter and substituting into the other, or by using trigonometric/hyperbolic identities.