1.1 Q-7

Question Statement

We are given the following parametric equations and need to show that they represent specific conic sections:

i. $x = a t^2$ , $y = 2 a t$ represents the equation of a parabola $y^2 = 4 a x$ .

ii. $x = a \cos \theta$ , $y = b \sin \theta$ represents the equation of an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ .

iii. $x = a \sec \theta$ , $y = b \tan \theta$ represents the equation of a hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ .

These parametric equations describe different conic sections:

A parabola is a curve defined by the equation $y^2 = 4 a x$ , which can be derived from a specific set of parametric equations.
An ellipse is a curve that satisfies the equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ .
A hyperbola is defined by the equation $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ .

We will Learn more about Conic sections in Chapter 6

Our goal is to manipulate the parametric equations given and show that they reduce to the standard forms of these conic sections.

We are given the parametric equations:

x = a t^2 \tag{1}

y = 2 a t \tag{2}

To find the equation of the parabola, we eliminate the parameter $t$ .

From equation (2), solve for $t$ :

t = \frac{y}{2a}

Now substitute this expression for $t$ into equation (1):

x = a \left(\frac{y}{2a}\right)^2 = a \cdot \frac{y^2}{4a^2} = \frac{y^2}{4a}

This simplifies to:

y^2 = 4 a x

Which is the equation of a parabola.

We are given the parametric equations:

x = a \cos \theta \tag{1}

y = b \sin \theta \tag{2}

To find the equation of the ellipse, we use trigonometric identities. First, solve for $\cos \theta$ and $\sin \theta$ from the parametric equations:

\cos \theta = \frac{x}{a} \tag{3}

\sin \theta = \frac{y}{b} \tag{4}

Now square both equations (3) and (4):

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

This is the standard equation of an ellipse:

\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1

We are given the parametric equations:

x = a \sec \theta \tag{1}

y = b \tan \theta \tag{2}

To find the equation of the hyperbola, we use trigonometric identities. First, solve for $\sec \theta$ and $\tan \theta$ :

\sec \theta = \frac{x}{a} \tag{3}

\tan \theta = \frac{y}{b} \tag{4}

Now square both equations (3) and (4):

\left(\frac{x}{a}\right)^2 - \left(\frac{y}{b}\right)^2 = \sec^2 \theta - \tan^2 \theta

Using the identity $\sec^2 \theta - \tan^2 \theta = 1$ , we get:

\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1

This is the standard equation of a hyperbola.

To eliminate the parameter $t$ or $\theta$ , use substitution and algebraic manipulation.
For the parabola, use the relationships between $x$ and $y$ expressed in terms of $t$ .
For the ellipse, use the identity $\cos^2 \theta + \sin^2 \theta = 1$ .
For the hyperbola, use the identity $\sec^2 \theta - \tan^2 \theta = 1$ .

For the parabola: Eliminate $t$ from the parametric equations and simplify to get $y^2 = 4 a x$ .
For the ellipse: Use trigonometric identities to square and add the parametric equations to get $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ .
For the hyperbola: Use trigonometric identities to square and subtract the parametric equations to get $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ .