A parabola is a fundamental concept in mathematics and geometry, categorized as one of the conic sections. It is formed by the intersection of a plane and a double-napped cone. The result is a U-shaped curve that can either open upwards or downwards, depending on its equation.
Parabolas appear in various real-world applications, from the trajectory of projectiles in physics to the design of satellite dishes and suspension bridges. They are also widely used across disciplines such as physics, engineering, finance, and computer science.
In this article, we will understand what a parabola is, its graph and properties, examples, and the Equation of a Parabola, and others in detail.
Definition and Key Elements
A parabola is a symmetrical curve that is defined as the set of all points that are equidistant from a fixed point (the focus) and a fixed straight line (the directrix). Both the focus and directrix play a vital role in determining the parabola's shape and position.
As one of the basic curves in conic sections, a parabola can be derived by slicing a double-napped cone (two cones stacked on top of each other at the vertex) with a plane at various angles. This geometrical relationship helps distinguish the parabola from other conic sections.
Parabola Shape
A parabola is a U-shaped curved line where every point on the line is at an equal distance from the focus and directrix of the parabola.
ParabolaEquation of Parabola
Equation of Parabola can vary depending on its orientation and the position of its vertex, but one common form is:
y = ax2 + bx + c
Here, a, b, and c are constants. The shape of the parabola depends primarily on the value of a:
- If a > 0, the parabola opens upwards.
- If a < 0, the parabola opens downwards.
Standard Equation of Parabola
Standard Equation of Parabola is given as follows:
y2 = 4ax
In this form, directrix is parallel to the y-axis.
If directrix is parallel to the x-axis, then the standard equation of a parabola is given by,
x2 = 4ay
If the parabolas are drawn in alternate quadrants, then their equation is given as y2 = -4ax and x2 = -4ay.
Form | y2 = 4ax | y2 = – 4ax | x2 = 4ay | x2 = – 4ay |
|---|
| Vertex | (0, 0) | (0,0) | (0, 0) | (0, 0) |
|---|
| Focus | (a, 0) | (-a, 0) | (0, a) | (0, -a) |
|---|
| Equation of Axis | y = 0 | y = 0 | x = 0 | x = 0 |
|---|
| Equation of Directrix | x = – a | x = a | y = – a | y = a |
|---|
| Length of Latus Rectum | 4a | 4a | 4a | 4a |
|---|
General equation of a parabola is given by y = a(x – h)2 + k or x = a(y – k)2 +h where (h, k) denotes the vertex.
(Regular form) y = a(x – h)2 + k
(Sidewise from) x = a(y – k)2 + h
Parametric Coordinates of a Parabola
For a parabola, y2 = 4ax, if we take x = at2 and y = 2at for any value of “t” they will satisfy the equation of a parabola, the coordinates (at2, 2at) is termed as parametric coordinate, and "t" is called as the parameter.
Thus, x = at2 and y = 2at are called the parametric equations of the parabola y2 = 4ax
Similarly, parametric form of the parabola x2 = 4ay are x = 2at, y = at2
Equation of Tangent to a Parabola
Tangents are lines that touch the curve only at a single point. So a line that touches the parabola exactly at one single point is called the tangent to a parabola.
There are various ways to find the tangent of a parabola which are discussed in next sections.
For the given parabola y2 = 4ax equation of the tangent at point (x1, y1) is given by:
yy1 = 2a(x+x1)
where,
(x1, y1) is the point of contact between the tangent and the curve.
For the given parabola y2 = 4ax equation of the tangent at point (at2, 2at) is given by:
ty = x + at2
where,
(at2, 2at) is the point of contact between the tangent and the curve.
For the given parabola y2 = 4ax with slope m equation of the tangent at point (a/m2, 2a/m) is given by
y = mx + a/m
where,
(a/m2, 2a/m) is the point of contact between the tangent and the curve.
Pair of Tangent from an External Point
Pair of tangents from an external point to any conic is given by SS1 = T2 where for parabola y2 = 4ax, S = y2 - 4ax, S1 = y12 -4ax1 and T = yy1 – 2a(x + x1).
Thus, the equation of pair of tangents from an external point becomes:
(y2 - 4ax)( y12 -4ax1) = [yy1 – 2a(x + x1)]2
Director Circle of Parabola
Director circle is the geometric object related to the conic section and is defined as the locus of the intersection of the pair perpendicular tangent of any conic. For the parabola, the director circle is the directrix as all the perpendicular pairs of tangents of the parabola intersect each other at the directrix.
Chord of contact of the parabola is a secant line joining the point of tangency for the tangents drawn from the external point on the parabola. For parabola y2 = 4ax, chord of contact is given by T = 0, where T = yy1 – 2a(x + x1). Therefore the equation of chord of contact is given
T = yy1 – 2a(x + x1) =0
Where, (x1, y1) is the external point from which both the tangents are drawn to the parabola.
Equation of Normal to a Parabola
A line perpendicular to the tangent of the parabola at the point of tangency is known as the normal of the parabola. As this line is perpendicular to the tangent at the point of tangency to the parabola, the equation of this line can be found easily if the equation of tangent and point of tangency is given, using the concept of the equation of line perpendicular to the given line, but this is not always the case.
The equation of normal is given in three ways as follows:
For a parabola y2 = 4ax and m is the slope of normal at the point of contact (am2, -2am), the equation of normal is given by:
y = mx – 2am – am3
For a parabola y2 = 4ax, equation of normal at (x1, y1) is given as follows:
y – y1 = (-y1/2a)(x – x1)
For a parabola y2 = 4ax, the equation of normal at the point (at2, 2at) [where t is the parameter] is given as follows:
y = -tx + 2at + at3
Some important parabola formulas are added in the table below:
| 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:
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.
- 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.
Solved Examples on Parabola
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 x 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:
Conclusion
Understanding the properties and applications of parabola 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.
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