Mastering the Standard Equations of a Parabola

Welcome to the foundational module on parabolas, a crucial topic in coordinate geometry for competitive exams like JEE. A parabola is a symmetrical open curve formed by the intersection of a cone with a plane parallel to its side. Understanding its standard equations is key to solving a wide range of problems involving curves, trajectories, and optimization.

What is a Parabola?

A parabola is the set of all points equidistant from a fixed point (focus) and a fixed line (directrix).

Imagine a point moving such that its distance to a fixed point (the focus) is always equal to its distance to a fixed line (the directrix). The path traced by this moving point forms a parabola.

Mathematically, a parabola is defined as the locus of points $(x, y)$ such that the distance from $(x, y)$ to the focus $F(a, 0)$ is equal to the distance from $(x, y)$ to the directrix $x = -a$ . This definition leads to the fundamental equation of a parabola.

Standard Equations of a Parabola

There are four primary standard forms of a parabola, each defined by its orientation and the position of its vertex and focus. These forms are essential for quickly identifying and analyzing parabolic curves.

Equation	Vertex	Focus	Directrix	Axis of Symmetry	Shape
$y^2 = 4ax$	(0, 0)	(a, 0)	$x = -a$	x-axis	Opens right
$y^2 = -4ax$	(0, 0)	(-a, 0)	$x = a$	x-axis	Opens left
$x^2 = 4ay$	(0, 0)	(0, a)	$y = -a$	y-axis	Opens up
$x^2 = -4ay$	(0, 0)	(0, -a)	$y = a$	y-axis	Opens down

The parameter 'a' in these equations represents the distance from the vertex to the focus (and also from the vertex to the directrix). A positive 'a' indicates the parabola opens in the positive direction of its axis (right for x-axis, up for y-axis), while a negative 'a' indicates opening in the negative direction.

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Parabolas with Vertex at (h, k)

When the vertex of the parabola is shifted from the origin (0,0) to a general point (h, k), the standard equations are modified by replacing 'x' with '(x-h)' and 'y' with '(y-k)'.

Equation	Vertex	Focus	Directrix
$(y-k)^2 = 4a(x-h)$	(h, k)	(h+a, k)	$x = h-a$
$(y-k)^2 = -4a(x-h)$	(h, k)	(h-a, k)	$x = h+a$
$(x-h)^2 = 4a(y-k)$	(h, k)	(h, k+a)	$y = k-a$
$(x-h)^2 = -4a(y-k)$	(h, k)	(h, k-a)	$y = k+a$

What is the directrix of the parabola given by the equation

(x-2)^2 = 8(y-3)

The equation is in the form $(x-h)^2 = 4a(y-k)$ . Here, $h=2$ , $k=3$ , and $4a=8$ , so $a=2$ . The directrix for this form is $y = k-a$ . Therefore, the directrix is $y = 3-2 = 1$ .

Key Properties and Applications

Parabolas have significant applications in physics and engineering. For instance, the trajectory of a projectile under gravity (neglecting air resistance) is parabolic. Also, the reflective property of a parabola is used in satellite dishes, telescopes, and headlights, where parallel rays converge at the focus.

Remember that the 'a' value determines the 'width' of the parabola. A larger absolute value of 'a' results in a narrower parabola, while a smaller absolute value results in a wider one.

If a parabola has its vertex at (1, 2) and opens upwards, and its focus is at (1, 4), what is its standard equation?

The vertex is (h, k) = (1, 2). Since it opens upwards, the form is $(x-h)^2 = 4a(y-k)$ . The focus is (h, k+a) = (1, 4). Thus, $k+a = 4$ , so $2+a = 4$ , which means $a=2$ . The equation is $(x-1)^2 = 4(2)(y-2)$ , or $(x-1)^2 = 8(y-2)$ .

Learning Resources

Conic Sections: The Parabola - Khan Academy(video)

Provides a clear, visual introduction to parabolas, their definition, and basic properties, suitable for building foundational understanding.

Parabola - Standard Equations and Properties - Byju's(documentation)

A comprehensive resource detailing the standard equations of parabolas, their derivations, and key properties with examples.

Conic Sections - Parabola - Maths is Fun(documentation)

Explains the concept of a parabola using simple language and interactive elements, making it accessible for learners.

JEE Mathematics: Conic Sections - Parabola - Vedantu(documentation)

Focuses on the JEE syllabus for parabolas, covering standard forms, properties, and common problem-solving techniques.

Understanding the Parabola: Focus, Directrix, and Vertex - YouTube(video)

A detailed video tutorial that visually explains the relationship between the focus, directrix, vertex, and the standard equation of a parabola.

Conic Sections - Parabola - Brilliant.org(documentation)

Offers an interactive approach to learning about parabolas, including their geometric definition and algebraic representation.

The Parabola: Definition, Equation, Properties & Applications - Toppr(documentation)

Covers the definition, standard equations, properties, and real-world applications of parabolas, with a focus on exam relevance.

Coordinate Geometry: Conic Sections - Parabola - Doubtnut(documentation)

Provides solved examples and explanations for parabola problems commonly encountered in competitive exams.

Parabola Properties and Equations - MathWorld(documentation)

A more advanced resource with rigorous mathematical definitions, properties, and related theorems concerning parabolas.

JEE Main 2024 Mathematics: Conic Sections - Parabola - Unacademy(blog)

A blog post offering tips and strategies for tackling parabola questions in JEE Main, including key formulas and problem-solving approaches.