A parabola in the conic section is referred to as an equation of a curve such that a location on the curve is equidistant from a fixed point, and a fixed-line. There are different sorts of conic sections in Mathematics like circles that can be defined based on the angle established between the plane and the intersection of the right circular cone with that for example parabola,ellipse and hyperbola.

With the article on the parabola, you will learn about the parabola definition, the parabolic equation, the formula of a parabola, the shape, derivation of the equation of a parabola, various applications relating the same with solved examples and more.

What is a parabola?

A conic section is the locus of a point that advances in such a way that its measure from a fixed point always exhibits a constant ratio to its perpendicular distance from a fixed position, all existing in the same plane. A parabola in conics is defined as the locus of a point that is equidistant from a fixed point named focus and from a fixed straight line named the directrix.

Standard Form of a Parabola

Conic sections are one of the important topics in Mathematics. A plane curve that is mirror-symmetrical and usually is of U shape is called a parabola in conics. The standard parabola forms of a regular parabola are as follows:

\(y^2=4ax\)

In this parabola form, the focus of the parabola lies on the positive side of the X−axis.

\(y^2=-4ax\)

In this parabola form, the focus of the parabola lies on the negative side of the X−axis.

\(x^2=4ay\)

In this parabola form, the focus of the parabola lies on the positive side of the Y−axis.

\(x^2=-4ay\)

Learn about the Section Formula in the linked article here!

Parabola Equation

In addition to the standard form of a parabola if the vertex of a parabola is at some point say A (h, k) and the length of the latus rectum is equal to p, then the general parabolic equations are:

\(y=p\ (x-h)^2+k \) is the regular form.

\(x=p\ (y-k)^2+h\) is the sidewise form.

Similarly if you want to learn about Equation of Ellipse, you can check it in the linked article.

Derivation of Equation of a Parabola

In the previous heading, we learn the standard and general equation of parabola. Now let us understand how we can derive the same. Consider the below image, here we have taken a point L(x, y) on the parabola.

In parabola eccentricity=1

I.e distance of a point from the focus /distance of this point from the directrix=1.

i.e. LF/LM=1

By using distance formula:

\(LF=\sqrt{\left(x-a\right)^2+\left(y-0\right)^2}\)

\(LF=\sqrt{\left(x-a\right)^2+\left(y\right)^2}\)

Similarly,

\(LM=\sqrt{\left(x+a\right)^2+\left(y-y\right)^2}\)

\(LM=\sqrt{\left(x+a\right)^2}\)

Earlier we saw LF/LM=1

LF=LM

\(\sqrt{\left(x-a\right)^2+\left(y\right)^2}=\sqrt{\left(x+a\right)^2}\)

Squaring on both side we get:

\(\left(x-a\right)^2+\left(y\right)^2=\left(x+a\right)^2\)

\(x^2 + a^2 – 2ax + y^2 = x^2 + a^2 + 2ax\)

\(y^2-2ax=2ax\)

\(y^2=4ax\)

Hence we have arrived at the equation. In the same manner, other equations can be derived.

Learn about Equation of Hyperbola and Equation of Hyperbola

Parabola Formula

We can state that when the axis of symmetry lies along the x-axis, the given parabola either opens to the left or the right depending on the coefficient value of x. Similarly, when the axis of symmetry lies along the y-axis, the given parabola either opens to the top or the bottom depending on the coefficient value of y. This implies that as per the x-axis and y-axis the formula varies.

For \(y^2=4ax\)

• Coordinates of vertex are (0, 0)
• Coordinates of focus is (a, 0)
• Equation of directrix is x + a=0
• Equation of axis is y = 0
• Length of Latus Rectum is 4a

Similarly for,

\(x^2=4ay\)

• Coordinates of vertex are (0, 0)
• Coordinates of focus is (0, a)
• Equation of directrix is y + a=0
• Equation of axis is x= 0
• Length of Latus Rectum is 4a

Learn about Parabola Ellipse and Hyperbola

Important Terms used in a Parabola

Axis: The straight line passing through the focus and perpendicular to the directrix is designated as the axis of the conic section.

Vertex: The point of intersection of a conic section and its axis is called the vertex of the conic section.

Focus: The point (a, 0) in the standard form image depicts the focus of the parabola.

Directrix: The lines formed parallel to the y-axis/x-axis and crossing through the point (-a, 0) or (0, a) or (a, 0) or (0, -a) is called the directrix of the parabola. The directrix is perpendicular to the axis of the parabola.

Focal Chord: The focal chord of a parabola is the chord progression by the focus of the parabola. The focal chord intersects the parabola at two distinct points.

Focal Distance: The distance of a point (x,y) on the parabola, from the focus, is the focal distance. The focal distance is equivalent to the perpendicular length of this point from the directrix.

Latus Rectum: Latus Rectum is the focal chord that is perpendicular to the axis of the parabola and passes within the focus of the parabola.

Eccentricity: The fixed ratio of the distance of point lying on the conics from the focus to its perpendicular distance from the directrix is termed the eccentricity of a conic section and is indicated by e.

For a parabola, the value of eccentricity is e = 1.

Applications of a Parabola

• Mirrors employed to focus light rays at a point are parabolic.
• The route is traversed by an object launched into the air and the stretched arc of a rocket launch is parabolic.
• Telescopes use parabolic mirrors.
• Parabolic mirrors are used in solar ovens to focus light beams for heating.
• Sound waves are focused on parabolic microphones.
• Car headlights, spotlights, automobile headlights are designed based on the parabola’s principles.
• The satellite dish used in satellites is a parabolic structure that provides focus and reflection of radio waves.

Also, read about Directrix of a Parabola, here

Solved Examples of Parabola

Some of the solved examples for better understanding of the topic in terms of formulas, equations and definitions are discussed below.

Question 1:Find the equation of the parabola with focus at F(3, 0) and directrix x = – 3 ?

Solution: Here, we have to find the equation of the parabola whose focus is at F(3, 0) and directrix x = – 3. As we know that, parabola of the form \(y^2 = 4ax\) has focus at (a, 0) and equation of directrix is given by x = – a

So, by comparing the focus F(3, 0) and directrix x = – 3 with (a, 0) and x = – a respectively we get

⇒ a = 3

So, the equation of the required parabola is \(y^2 = 4 ⋅ 3 ⋅ x = 12x\)

Question 2: Find the coordinates of focus of the parabola \(x^2 = 32y\).

Solution:

The focus of the parabola \(x^2 = 4ay\) is given by: (0, a)

Given, \(x^2 = 32y\)

The given equation can be written as \(x^2 = 4 ⋅ 8 ⋅ y\)

By comparing the above equation with the standard equation of the parabola, \(x^2 = 4ay\)

⇒ a = 8

As we know that, the focus of the parabola \(x^2 = 4ay\) is given by: (0, a)

⇒ (0, a) = (0, 8).

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