Mastering Parametric Equations of a Parabola for Competitive Exams

Welcome to this module on Parametric Equations of a Parabola, a crucial topic for competitive exams like JEE. Parametric equations offer a powerful way to describe curves by expressing the coordinates (x, y) of points on the curve as functions of a third variable, known as the parameter. For a parabola, this parameter often simplifies calculations and provides elegant solutions to various problems.

Understanding the Standard Parabola and its Parametric Form

The standard equation of a parabola with its vertex at the origin and opening to the right is $y^2 = 4ax$ . While this form is useful, it can be cumbersome for certain calculations. Parametric equations provide an alternative representation that is often more manageable.

The standard parametric form of a parabola $y^2 = 4ax$ is $(at^2, 2at)$.

For a parabola defined by $y^2 = 4ax$ , any point on the parabola can be represented by the coordinates $(at^2, 2at)$ , where 't' is the parameter. This form is derived by letting $y = 2at$ , which then satisfies the equation $y^2 = 4ax$ when $x = at^2$ .

Consider the standard parabola $y^2 = 4ax$ . If we let $y = 2at$ , where 'a' is a constant related to the focal length and 't' is the parameter, we can substitute this into the equation: $(2at)^2 = 4ax$ . This simplifies to $4a^2t^2 = 4ax$ . Dividing both sides by $4a$ (assuming $a eq 0$ ), we get $at^2 = x$ . Thus, any point on the parabola $y^2 = 4ax$ can be represented parametrically as $(at^2, 2at)$ . The parameter 't' can take any real value, and each value of 't' corresponds to a unique point on the parabola.

Parametric Forms for Other Orientations

The parametric form can be adapted for parabolas with different orientations and vertex positions.

Parabola Equation	Parametric Form (x, y)	Parameter Range
$y^2 = 4ax$	$(at^2, 2at)$	$t \in \mathbb{R}$
$y^2 = -4ax$	( $-at^2, 2at$ )	$t \in \mathbb{R}$
$x^2 = 4ay$	$(2at, at^2)$	$t \in \mathbb{R}$
$x^2 = -4ay$	$(2at, -at^2)$	$t \in \mathbb{R}$
$(y-k)^2 = 4a(x-h)$	$(h + at^2, k + 2at)$	$t \in \mathbb{R}$

Key Properties and Applications in Parametric Form

Parametric equations simplify the derivation and application of various properties of the parabola, such as tangents, normals, and chords.

What is the parametric form of a parabola with vertex at

(h, k)

and equation

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

The parametric form is $(h + at^2, k + 2at)$ .

Consider the tangent to the parabola $y^2 = 4ax$ at a point $P(at^2, 2at)$ . The slope of the tangent is found by differentiating $y = \pm 2\sqrt{ax}$ with respect to $x$ . Using the parametric form, $\frac{dx}{dt} = 2at$ and $\frac{dy}{dt} = 2a$ . Therefore, $\frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{2a}{2at} = \frac{1}{t}$ . The equation of the tangent at $P(at^2, 2at)$ is $y - 2at = \frac{1}{t}(x - at^2)$ , which simplifies to $yt = x + at^2$ .

The parametric representation of a parabola $y^2 = 4ax$ as $(at^2, 2at)$ is a fundamental concept. The parameter 't' can be visualized as related to the slope of the line connecting the origin to the point on the parabola, or more directly, as related to the angle a line from the focus makes with the axis of symmetry. For instance, if we consider a line from the focus $(a, 0)$ to a point $(at^2, 2at)$ , the slope of this line is $\frac{2at - 0}{at^2 - a} = \frac{2at}{a(t^2 - 1)} = \frac{2t}{t^2 - 1}$ . The slope of the tangent at $(at^2, 2at)$ is $1/t$ . This relationship between the parameter and geometric properties is key to solving problems efficiently.

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Parametric equations are particularly useful for problems involving chords of contact, intersection of tangents, and locus problems related to parabolas.

Practice Problems and Strategies

When solving problems involving parametric equations of parabolas, always identify the correct parametric form based on the given equation. Substitute the parametric coordinates into the conditions of the problem and solve for the parameter 't' or related quantities. Remember that 't' is a real number, and its range might be restricted in specific problem contexts.

What is the slope of the tangent to the parabola

y^2 = 4ax

at the point corresponding to parameter 't'?

The slope is $1/t$ .

For example, to find the equation of the tangent to $y^2 = 12x$ at the point where $t=2$ , we first identify $4a = 12$ , so $a=3$ . The point is $(3 \cdot 2^2, 2 \cdot 3 \cdot 2) = (12, 12)$ . The slope is $1/t = 1/2$ . The tangent equation is $y - 12 = \frac{1}{2}(x - 12)$ , which simplifies to $2y - 24 = x - 12$ , or $x - 2y + 12 = 0$ .

Learning Resources

Parametric Equations of Conic Sections - Brilliant.org(documentation)

Provides a clear explanation of parametric equations for various conic sections, including parabolas, with examples.

Parametric Equations of a Parabola - Maths is Fun(documentation)

A beginner-friendly explanation of parametric equations for parabolas, focusing on the standard form and its derivation.

Parametric Equations of a Parabola - YouTube Tutorial(video)

A visual tutorial demonstrating how to derive and use parametric equations for parabolas, often covering problem-solving techniques.

Conic Sections: Parametric Equations - Khan Academy(video)

Khan Academy's comprehensive video series on parametric equations, with a specific focus on parabolas and their properties.

Parametric Equations of a Parabola - Byju's(blog)

Offers a detailed explanation of parametric equations for parabolas, including standard forms, properties, and solved examples relevant to exams.

Parametric Equations of Conic Sections - Tutorialspoint(documentation)

Covers parametric representations of all conic sections, providing formulas and examples for parabolas.

JEE Mathematics: Conic Sections - Parametric Equations(blog)

A resource tailored for JEE aspirants, explaining parametric equations of parabolas with a focus on exam-oriented problems and strategies.

Parametric Equations of a Parabola - MathWorld(documentation)

Provides rigorous mathematical definitions, formulas, and properties of parabolas, including their parametric forms.

Problems on Parametric Equations of Parabola - Doubtnut(blog)

A platform with numerous solved problems and video explanations for various topics, including parametric equations of parabolas.

Parametric Equations - Wikipedia(wikipedia)

A general overview of parametric equations, their history, and applications across different fields of mathematics and science.