Parabola
Parabola is an important curve of the conic section of the coordinate geometry. Many physical motions of bodies follow a curvilinear path which is in the share of a parabola. A parabola is generally symmetric across one of the axis and has a focus and a directrix. A parabola as per definition is the locus of a point which is equidistant from a fixed point called the focus, and a fixedline called the directrix.
Here we shall aim at understanding the derivation of the standard formula of a parabola, the different standard forms of a parabola, and the formulas, properties of a parabola.
1.  Parabola  Definition 
2.  Derivation  Parabola Equation 
3.  Standard Equations of a Parabola 
4.  Properties of a Parabola 
5.  Solved Examples 
6.  Practice Questions 
7.  FAQs on Parabola 
Parabola  Definition
A parabola refers to an equation of a curve, such that a point on the curve is equidistant from a fixed point, and a fixedline. The fixed point is called the focus of the parabola, and the fixed line is called the directrix of the parabola. Also an important point to note is that the fixed point does not lie on the fixedline.
The standard equation of a parabola is y^{2} = 4ax
Some of the important terms below are helpful to understand the features and properties of a parabola.
 Focus: The point (a, 0) is the focus of the parabola
 Directrix: The line drawn parallel to the yaxis and passing through the point (a, 0) is 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 passing through the focus of the parabola. The focal chord cuts the parabola at two distinct points.
 Focal Distance: The distance of a point \((x_1, y_1)\) on the parabola, from the focus, is the focal distance. The focal distance is also equal to the perpendicular distance of this point from the directrix.
 Latus Rectum: It is the focal chord that is perpendicular to the axis of the parabola and is passing through the focus of the parabola. The length of the latus rectum is taken as LL' = 4a. The endpoints of the latus rectum are (a, 2a), (a, 2a).
 Eccentricity: (e = 1). It is the ratio of the distance of a point from the focus, to the distance of the point from the directrix. The eccentricity of a parabola is equal to 1.
Derivation  Parabola Equation
Let us consider a point P with coordinates (x, y) on the parabola. As per the definition of a parabola, the distance of this point from the focus F is equal to the distance of this point P from the Directrix. Here we consider a point M on the directrix, and the perpendicular distance PM is taken for calculations.
As per this definition, we have PF = PM
The coordinates of the focus is F(a,0) and we can use the coordinate distance formula to find its distance from P(x, y)
PF = \(\sqrt{(x  a)^2 + (y  0)^2}\)
= \(\sqrt{(x  a)^2 + y^2}\)
The equation of the directtrix is x + a = 0 and we use the perpendicular distance formula to find PM.
PM = \(\frac{x + a}{\sqrt{1^2 + 0^2}}\)
= x + a
We need to derive the equation of parabola using PF = PM
\(\sqrt{(x  a)^2 + y^2}\) = x + a
Squaring the equation on both sides,
(x  a)^{2} + y^{2} = (x + a)^{2}
x^{2} + a^{2}  2ax + y^{2} = x^{2} + a^{2} + 2ax
y^{2}  2ax = 2ax
y^{2 }= 4ax
Now we have successfully derived the standard equation of a parabola.
Standard Equations of a Parabola
There are four standard equations of a parabola. The four standard forms are based on the axis of the parabola, and the orientation of the parabola. The transverse axis and the conjugate axis of each of these parabolas are different. The below image presents the four standard equations and graphs of a parabola.
Let us now look at the properties and formulas of each of the standard equations of a parabola. The best method of learning these formulas is to compare and understand the transformation and relationship across the formulas.
Forms of Parabola:  y^{2} = 4ax  y^{2} = 4ax  x^{2} = 4ay  x^{2} = 4ay 
Equation of axis:  y = 0  y = 0  x = 0  x = 0 
Equation of the directrix:  x + a = 0  x  a = 0  y + a = 0  y  a = 0 
Vertex  (0, 0)  (0, 0)  (0, 0)  (0, 0) 
Focus  (a, 0)  (a, 0)  (0, a)  (0, a) 
Length of Latus Rectum  4a  4a  4a  4a 
Properties of a Parabola
Here we shall aim at understanding some of the important properties and terms related to a parabola.
Tangent: The tangent is a line touching the parabola. The equation of a tangent to the parabola y^{2} = 4ax at the point of contact \((x_1, y_1)\) is \(yy_1 = 2a(x + x_1)\).
Normal: The line drawn perpendicular to tangent and passing through the point of contact and the focus of the parabola is called the normal. For a parabola y^{2} = 4ax, the equation of the normal passing through the point \((x_1, y_1)\) and having a slope of m = y1/2a, the equation of the normal is \((y  y_1) = \dfrac{y_1}{2a}(x  x_1)\)
Chord of Contact: The chord drawn to joining the point of contact of the tangents drawn from an external point to the parabola is called the chord of contact. For a point \((x_1, y_1)\) outside the parabola, the equation of the chord of contact is \(yy_1 = 2x(x + x_1)\).
Pole and Polar: For a point lying outside the parabola, the locus of the points of intersection of the tangents, draw at the ends of the chords, drawn from this point is called the polar. And this referred point is called the pole. For a pole having the coordinates \((x_1, y_1)\), for a parabola y^{2} =4ax, the equation of the polar is \(yy_1 = 2x(x + x_1)\).
Parametric Coordinates: The parametric coordinates of the equation of a parabola y^{2} = 4ax are (at^{2}, 2at). The parametric coordinates represent all the points on the parabola.
Solved Examples on Parabola

Example 1: The equation of a parabola is y^{2 }= 24x. Find its length of latus rectum, focus, and vertex.
Solution:
To find: Length of latus rectum, focus and vertex of the parabola
Given: Equation of a parabola: y^{2 }= 24x
Therefore, 4a = 24
a = 24/4 = 6
Now, parabola formula for latus rectum is:
Length of latus rectum = 4a
= 4(6)
= 24
Now, parabola formula for focus:
Focus = (a,0)
= (6,0)
Now, parabola formula for vertex:
Vertex = (0,0)
Answer: Length of latus rectum = 24, focus = (6,0), vertex = (0,0)

Example 2: The equation of a parabola is y^{2} = 8x. Find its length of latus rectum, focus, and vertex.
Solution:
To find: Length of latus rectum, focus and vertex of a parabola
Given: Equation of a parabola: y^{2 } = 8x
On comparing it with the general equation of a parabola y^{2}= 4ax, we get
4a = 8, and a = 2
Length of latus rectum = 4a = 4(2) = 8
Focus = (0,a) = (0,2)
Vertex = (0,0)
Answer: Length of latus rectum = 8, focus = (0, 2), vertex = (0, 0).
FAQs on Parabola
What is Parabola in Conic Section?
Parabola is an important curve of the conic section. It is the locus of a point that is equidistant from a fixed point, called the focus, and the fixedline called the directrix. Many of the motions in the physical world follow a parabolic path. Hence learning the properties and applications of a parabola is the foundation for physicists.
What Is the Equation of Parabola?
The standard equation of a parabola is y^{2} = 4ax. The axis of the parabola is the xaxis which is also the transverse axis of the parabola. The focus of the parabola is F(a, 0), and the equation of the directrix of this parabola is x + a = 0.
What Is the Vertex of the Parabola?
The vertex of the parabola is the point where the parabola cuts through the axis. The vertex of the parabola having the equation y^{2} = 4ax is (0,0), as it cuts the axis at the origin.
How to Find Equation of a Parabola?
The equation of the parabola can be derived from the basic definition of the parabola. A parabola is the locus of a point that is equidistant from a fixed point called the focus (F), and the fixedline called the Directrix (x + a = 0). Let us consider a point P(x, y) on the parabola, and using the formula PF = PM, we can find the equation of the parabola. Here the point 'M' is the foot of the perpendicular from the point P, on the directrix. Hence, the derived standard equation of the parabola is y^{2} = 4ax.
What is the Eccentricity of Parabola?
The eccentricity of a parabola is equal to 1 (e = 1). The eccentricity of a parabola is the ratio of the distance of the point from the focus to the distance of this point from the directrix of the parabola.
What Is the Foci of a Parabola?
The parabola has only one focus. For a standard equation of the parabola y^{2} = 4ax, the focus of the parabola is F(a, 0). It is a point lying on the xais and on the transverse axis of the parabola.
What Is the Conjugate Axis of a Parabola?
The line perpendicular to the transverse axis of the parabola and is passing through the vertex of the parabola is called the conjugate axis of the parabola. For a parabola y^{2} = 4ax, the conjugate axis is the yaxis.
What Are the Vertices of a Parabola?
The point on the axis where the parabola cuts through the axis is the vertex of the parabola. The vertex of the parabola for a standard equation of a parabola y^{2} = 4ax is equal to (0, 0). The parabola cuts the xaxis at the origin.
How to Find Transverse Axis of a Parabola?
The line passing through the vertex and the focus of the parabola is the transverse axis of the parabola. The standard equation of the parabola y^{2} = 4ax has the xaxis as the axis of the parabola.