Formula/Property	Equation or Description
Standard Equation of Parabola	y2 = 4ax (or x2 = 4ay for a different orientation)
Focus	(a, 0) for y2 = 4ax (or (0, a) for x2 = 4ay)
Directrix	x = -a for y2 = 4ax (or y = -a for x2 = 4ay)
Latus Rectum	4a
Eccentricity	1
General Equations of Parabola	y = a(x – h)2 + k OR x = a(y – k)2 + h
Parametric Equation of Parabola	x = 2at, y = at2
Equation of Tangent	yy1 = 2a(x + x1)
Equation of Tangent in Parametric Form	ty = x + at2 [where, (at2, 2at) is the point of contact]
Equation of Tangent in Slope Form	y = mx + a/m  [where m is the slope of tangent]
Pair of Tangent from an External Point	(y2 – 4ax)( y12 -4ax1) = [yy1 – 2a(x + x1)]2
Director Circle	Directrix i.e., x = -a [For y2 = 4ax]
Chord of Contact	yy1 – 2a(x + x1) =0
Equation of Normal in Slope Form	y = mx – 2am – am3
Equation of Normal in Normal Form	y – y1 = (-y1/2a)(x – x1)
Equation of Normal in Parametric Form	y = -tx + 2at + at3

Articles Related to Parabola:

• Equation of Circle
• Equation of Ellipse
• Equation of Hyperbola

Derivation of Parabola Equation

Take a point P with coordinates (x, y) on the parabola which lies on the X-Y plane. By the definition of the parabola, the distance of any point on the parabola from the focus and from the directrix is equal.

Now distance of P from the directrix is given by PB where the coordinates of B are (-a, y) as it lies on the directrix, and the distance of P from focus is PF.

Image of parabola is shown below,

By the definition of parabola, PF = PB . . . .(1)

Using Distance Formula, we get

PF = √(x-a)²+(y-0)²= √{(x-a)²+y²} . . . .(2)

PB = √{(x+a)²} . . . .(3)

By using, equations (1), (2), and (3), we get

√{(x-a)²+y²} = √{(x+a)²}

⇒ (x – a)² + y² = (x + a)²

⇒ x² + a² – 2ax + y² = x² + a² + 2ax

⇒ y² – 2ax = 2ax

y² = 4ax

Which is the required equation of the parabola.

Similarly, the equation for other parabolas i.e., x² = 4ay,  y² = -4ax, and x² = -4ay, can also be proved. 

Graph of Parabola

Graph of the parabola is a U-shaped curve, which can open either in an upward direction or in a downward direction. Generally, the equation of a parabola which is graphed is written in the form of y = ax² + bx + c, where a, b, and c are constants that define the shape of the parabola.

If a > 0, in the above equation, the parabola opens in an upward direction and its vertex is the lowest point of the parabola, and if a<0, then the parabola opens in a downward direction and its vertex is the highest point in the parabola. Vertex of the parabola is also the point from where the only line of symmetry of the parabola passes.

Position of Point Relative to the Parabola

Position of a point A (x₁, y₁) relative to the parabola y² = 4ax, can be shown using the S₁ = y² – 4ax,

Case 1: If S₁ = 0, for any point A, then Point A lies on the parabola.

Case 2: If S₁ < 0, for any point A, then Point A lies inside the parabola.

Case 3:  If S₁ > 0, for any point A, then Point A lies outside the parabola.

Intersection with Straight Line

For a parabola y² = 4ax, any straight-line y = mx + c, can almost intersect the parabola at two points. For the intersection of the line and parabola, put y = mx + c in the equation of the parabola,

(mx + c)² = 4ax

⇒ m²x² + c² + 2mxc = 4ax

⇒ m²x² + (2mc – 4a)x + c²= 0

⇒ Discriminant = (2mc – 4a)² – 4m²c²

Case 1: Discriminant > 0:

For positive discriminant, quadratic equations have two real solution,

⇒ There are two points of intersection between line and parabola.

Case 2: Discriminant = 0:

For discriminant equal to 0, quadratic equations have only real solution (common root),

⇒ There are open point of intersection between line and parabola i.e., line is tangent to parabola.

Case 3: Discriminant < 0:

For negative discriminant, quadratic equations have no real roots,

⇒ There are no point of intersection between line and parabola.

Properties of Parabola

• Axis of Symmetry: Line that is perpendicular to the directrix and passes through the focus is called the axis of symmetry. The parabola is symmetrical about its axis of symmetry.
• Vertex: Point where the parabola intersects its axis of symmetry is called the vertex. The vertex is the point where the parabola changes direction from opening upwards to opening downwards (or vice versa).
• Focal Length: Distance between the vertex and the focus is called the focal length. All parabolas with the same focal length are similar.
• Directrix: Fixed line from which any point on the parabola is the same distance as the focus.
• Reflective Property: If a parabola is made of a material that reflects light, then light that travels parallel to the axis of symmetry of a parabola and strikes its concave side is reflected to its focus, regardless of where on the parabola the reflection occurs. Conversely, light that originates from a point source at the focus is reflected into a parallel (“collimated”) beam, leaving the parabola parallel to the axis of symmetry.
• Equation of Parabola: Equation of a parabola depends on its orientation and position. The standard equation of a parabola that opens upwards and is centered at the origin is y = x². The standard equation of a parabola that opens downwards and is centered at the origin is y = -x².

Important Terms Related to Parabola

• Focus: Point (a, 0) is called the focus of a Standard Parabola (y² = 4ax), and it has a very special property that has various real-life applications i.e. if any light ray traveling parallel to the axis of the parabola, the parabola converges those light rays at the focus.

• Directrix: A line drawn perpendicular to the base axis and passing through the point (-a, 0) is the directrix of the parabola. Directrix is always perpendicular to the axis of a parabola.

• Focal Chord: A chord passing through the focus of the parabola is called the focal chord of a parabola. The focal chord always cuts the parabola at two distinct points.

• Focal Distance: Distance of any point (x₁,y₁) lying on the parabola, from the focus of the parabola, is called the focal distance. The focal distance equals the perpendicular distance of the given point from the directrix.

• Latus Rectum: A chord perpendicular to the axis of the parabola and passing through the focus of the parabola is called the Latus rectum. Points at the ending of the latus rectum are (a, 2a), (a, -2a), and their length is taken as LL’ = 4a.

• Eccentricity: Ratio of the distance of a point from the focus, and the distance of the point from the directrix is called eccentricity. Eccentricity for a parabola is 1.

Parabola Solved Examples

Example 1: Find coordinates of the focus, axis, the equation of the directrix, and latus rectum of the parabola y² = 16x.

Solution:

Given equation of the parabola is: y² = 16x

Comparing with the standard form y² = 4ax,

4a = 16 ⇒ a = 4

The coefficient of x is positive so the parabola opens to the right.

Also, the axis of symmetry is along the positive x-axis.

Therefore,

Focus of the parabola is (a, 0) = (4, 0).

Equation of the directrix is x = -a, i.e. x = -4 

Length of the latus rectum = 4a = 4(4) = 16

Example 2: Find the equation of the parabola which is symmetric about the y-axis, and passes through the point (3, -4).

Solution:

Given that parabola is symmetric about the y-axis and has its vertex at the origin.

Thus, equation can be of the form x² = 4ay or x² = -4ay, where the sign depends on whether the parabola opens upwards or downwards.

Since parabola passes through (3, -4) which lies in the fourth quadrant, it must open downwards.

So, equation will be: x² = -4ay

Substituting (3, -4) in the above equation,

(3)² = -4a(-4)

9 = 16a

a = 9/16

Hence, the equation of the parabola is: x² = -4(9/16)y 

                                                            4x² = -9y

Example 3: Find coordinates of the focus, axis, and the equation of the directrix and latus rectum of the parabola y² = 8x.

Solution:

Given equation of the parabola is: y² = 8x

Comparing with the standard form y² = 4ax,

4a = 8

a = 2

The coefficient of x is positive so the parabola opens to the right.

Also, the axis of symmetry is along the positive x-axis.

Therefore,

Focus of the parabola is (a, 0) = (2, 0).

Equation of the directrix is x = -a, i.e. x = -2

Length of the latus rectum = 4a = 4(2) = 8

Example 4: Find the coordinates of the focus, axis, the equation of the directrix, and latus rectum of the parabola y² = 52x.

Solution:

Given equation of parabola is: y² = 52x

Comparing with the standard form y² = 4ax,

4a = 52

a = 13

The coefficient of x is positive so the parabola opens to the right.

Also, the axis of symmetry is along the positive x-axis.

Therefore,

Focus of the parabola is (a, 0) = (13, 0).

Equation of the directrix is x = -a, i.e. x = -13

Length of the latus rectum = 4a = 4(13) = 52

Example 5: Find coordinates of the focus, axis, the equation of the directrix, and latus rectum of the parabola x² = 16y.

Solution:

Given equation of parabola is: x² = 16y

Comparing with standard form x² = 4ay,

4a = 16

a = 4

The coefficient of x is positive so the parabola opens upward.

Also, the axis of symmetry is along the positive x-axis.

Therefore,

Focus of the parabola is (0,a) = (0, 4).

Equation of the directrix is y= -a, i.e. y = -4

Length of the latus rectum = 4a = 4(4) = 16

Practice Questions on Parabola

Q1. Find the vertex, focus, and directrix of the parabola with the equation y = x² – 4x + 3y = x² – 4x + 3.

Q2. Determine whether the parabola with the equation y = -2x² + 4x – 1y = -2x² + 4x – 1 opens upward or downward, and find its vertex.

Q3. Given the equation 4x² – 16y = 0, 4x² – 16y = 0, rewrite it in standard form and find the vertex, focus, and directrix of the parabola.

Q4. Solve for xx in the equation 2x² – 3x – 5 = 0, 2x² – 3x – 5 = 0, and determine the nature of the roots with respect to the corresponding parabola.

Also Read:

• Ellipse
• Hyperbola
• Standard Equation of a Parabola
• Applications of Parabola in Real-Life
• Equation of Parabola from its Focus and Directrix

Conclusion

Understanding the properties and applications of parabolas is essential in both theoretical and applied mathematics. Their unique geometric characteristics and reflective properties make them a fascinating topic with wide-ranging applications in various fields. Whether grappling with parabola motion, designing architectural elements, or studying the nature of conic sections, the parabola remains a key concept.

A parabola is a fundamental concept in mathematics, particularly in the study of quadratic functions and conic sections. Its distinct U-shaped curve is defined by a quadratic equation and exhibits unique properties such as symmetry and a focal point, which have wide-ranging applications in various fields.

Parabola in Maths – FAQs

What is Parabola?

Parabola is the set of all points that are equidistant from a fixed point (called the focus) and a fixed straight line (called the directrix).

What is meant by Conjugate Axis of a Parabola?

A line that passes through the vertex of the parabola and is perpendicular to its transverse axis is called the conjugate axis of the parabola.

What are Applications of Parabola?

Parabolas are used for a variety of purposes some of them are

• Parabolic arch is used in the construction of various monuments.
• Parabolic mirrors are used in reflecting telescopes, satellites, etc.
• Parabolas are used in various mathematical calculations, such as tracing the path of a missile, the trajectory of a bullet, etc.

What is Shape of a Parabola?

Graph of a parabola is in the shape of U.

What is Eccentricity of a Parabola?

Eccentricity is defined as the ratio of distances of any point of conic section to its focus and corresponding directrix. For parabola eccentricity is 1.

What is Formula for Length of Latus Rectum of a Parabola?

For parabola y² = 4ax, length of the latus rectum is calculated by 4a..

What is Vertex of a Parabola?

Point of intersection of the parabola and both the conjugate axis is called the vertex of a parabola. If the equation of parabola is y² = 4ax, then the vertex is (0, 0).

What is a Parabolic motion?

Parabola motion, also known as projectile motion, describes the path followed by an object that is subject only to the force of gravity and no other external forces, such as air resistance. In this type of motion, the object moves in a curved path that resembles a parabola.

What are 4 Types of Parabola?

Parabolas can generally be categorized into four types based on their orientation and direction:

• Vertical Opening Up: Standard form is y = ax² + bx + c, where a > 0.

• Vertical Opening Down: Standard form is y = ax²+ bx + c, where a < 0.

• Horizontal Opening Right: Standard form is x = ay²+ by + c, where a > 0.

• Horizontal Opening Left: Standard form is x = ay²+ by + c, where a < 0.

What is difference between Parabola and Hyperbola?

Parabola is a U-shaped curve where any point is at an equal distance from a fixed point (the focus) and a fixed straight line (the directrix). It has one axis of symmetry.

Hyperbola is a curve where the difference of the distances from two points (the foci) is constant. It has two disconnected parts and two axes of symmetry.

What are 4 key Features of a Parabola?

• Vertex: Highest or lowest point of the parabola.

• Axis of Symmetry: Line that divides the parabola into two symmetric halves.

• Focus: A fixed point through which all the parallel light rays reflect off a parabolic mirror.

• Directrix: A fixed line that is equidistant from all points on the parabola.

Who invented Parabola?

Concept of the parabola and its mathematical properties have been studied and developed over centuries by various mathematicians and scholars. However, it was the ancient Greek mathematician Apollonius of Perga who extensively studied conic sections, including the parabola, around the 3rd century BCE.