Equation of Hyperbola Parametric Form, Tangents & Normals Explained

Hyperbola is a conic section developed by joining a right circular cone with a plane at an angle such that both halves of the cone are joined. This intersection creates two separate unbounded curves that are mirror images of each other. Conic sections form an integral part of analytical geometry like Parabola, Ellipse and Hyperbola. Like that of an ellipse, the hyperbola can also be interpreted as a set of points in the coordinate plane. Set of all points (x,y) in a plane such that the difference of the lengths between (x,y) and the foci is a positive constant is the hyperbola definition.

A hyperbola is the set of all points such that the difference between the distances to two fixed points (called foci) is always the same.
In other words, if you take any point on the hyperbola, measure how far it is from both foci, and subtract the shorter distance from the longer one — the answer will always be the same number.

For a point (x, y) on the hyperbola and for two foci(−c,0) and (c,0), the locus of the hyperbola is

where is the distance from (−c,0) to (x,y) and is the distance from (c,0) to (x,y).

What is the Equation of Hyperbola?

Standard equation of hyperbola with center (0,0) and transverse axis on the x-axis and the conjugate axis is the y-axis is as shown:

Form:

In this form of hyperbola, the center is located at the origin and foci are on the X-axis.

Standard equation of a hyperbola with center (0,0) and transverse axis on the y-axis and its conjugate axis is the x-axis is as shown:

Form:

In this form of hyperbola, the center is located at the origin and foci are on the Y-axis.

Standard equation of a hyperbola with center (h,k) and transverse axis parallel to the x-axis is as shown:

Standard equation of a hyperbola with center (h,k) and transverse axis parallel to the y-axis is as shown:

For the hyperbola

The parametric equation is and parametric coordinates of the point resting on it are presented by

Equation of Tangents and Normals to Hyperbola

A hyperbola is the set of all locations in a plane, the difference of whose lengths from two fixed locations in the plane is constant. ‘Difference’ here means the distance to the ‘farther’ position minus the distance to the ‘closer’ point. The two fixed points are the foci and the mid-point of the line segment connecting the foci is the center of the hyperbola.

Equation of a tangent to the hyperbola: in Point form:

At the point equation of a tangent is given by:

Slope Form: Equation of a tangent to hyperbola in terms of slope m:

Parametric Form: In parametric coordinates, the equation of the tangent is given as

Equation of normal to the hyperbola: in Point form:

At the point equation of normal is given by:

Slope Form: Equation of normal to hyperbola in terms of slope m:

Parametric Form: In parametric coordinates, the equation of the normal is given as

Learn about Equation of Parabola

Hyperbola Formula Table

Check out the hyperbola formula table given below:

Hyperbola Equation
Coordinates of the center	(0, 0)	(0, 0)
Coordinates of the Vertices	(a, 0) and (-a, 0)	(0, a) and (0, – a)
Coordinates of the foci	(c, 0) and (-c, 0)	(0, c) and (0, -c)
Length of the transverse axis	2a	2a
Length of the conjugate axis	2b	2b
The formula for the eccentricity of a hyperbola
latus rectum of hyperbola

Learn about Parabola Ellipse and Hyperbola

Hyperbola Graph

All hyperbolas share general features, consisting of two curves, each along with a vertex and a focus. The transverse axis of a hyperbola is the axis that passes through both vertices and foci, and the conjugate axis of the hyperbola is perpendicular to the transverse axis.

We can recognize the hyperbola graph in standard forms as shown below. If the equation of the provided hyperbola is not in standard form, then we require to complete the square to receive it in standard form.

Now observe the below graphs where the center is not zero. We can also recognize the different parts of a hyperbola in the hyperbola graphs.

Standard equation of a hyperbola with center (h,k) and transverse axis parallel to the x-axis is as shown:

• The coordinates of the center are (h, k).
• The coordinates of vertices are (h+a, k) and (h – a,k).
• The Co-vertices resemble “b” and the coordinates of co-vertices are (h,k+b) and (h,k-b).
• Foci possess the coordinates (h+c,k) and (h-c,k). The value of c is given as,
• The equations of the asymptotes are

Standard equation of a hyperbola with center (h,k) and transverse axis parallel to the y-axis is as shown:

• The coordinates of the center are (h, k).
• The coordinates of vertices are (h, k+a) and (h,k- a).
• The Co-vertices resemble “b” and the coordinates of co-vertices are (h+b,k) and (h-b,k).
• Foci possess the coordinates (h,k+c) and (h,k-c). The value of c is given as,
• The equations of the asymptotes are

Learn about Equation of Ellipse

Properties of Hyperbola

The next is the important properties associated with different concepts that would help in understanding hyperbola in a much better way.

Asymptotes: The pair of straight lines traced parallel to the hyperbola and thought to meet the hyperbola at infinity. The equations of the asymptotes of the hyperbola are as follows:

For

The equations of the asymptotes are

For

The equations of the asymptotes are

Rectangular Hyperbola: The hyperbola possessing the transverse axis and the conjugate axis of equal length is termed the rectangular hyperbola. Then we have 2a = 2b, or a = b. Therefore the equation of the rectangular hyperbola is equivalent to OR

Parametric Coordinates: The points on the hyperbola can be expressed with the parametric coordinates as follows:

For the hyperbola

The parametric equation is and parametric coordinates of the point resting on it are presented by

These parametric coordinates interpreting the points on the hyperbola satisfy the equation of the hyperbola.

Auxiliary circle of a hyperbola: A circle formed by the endpoints of the transverse axis of the hyperbolic curve as its diameter is named the auxiliary circle of a hyperbola. The equation of the auxiliary circle of the hyperbola is

Direction Circle: The locus of the point of intersection of perpendicular tangents to the hyperbola is termed the director circle. The equation of the director circle of the hyperbola is given as

Conjugate Hyperbola: Two hyperbolas such that the transverse axis and conjugate axis of one hyperbola are sequentially the conjugate axis and transverse axis of the other are termed conjugate hyperbola of one another.

are conjugate hyperbolas of each other.

Real-Life Applications of Hyperbola

1. Guitar Shape
   The narrow middle part of a classical guitar, called the waist, is shaped like a hyperbola. This shape not only looks attractive but also helps the guitar rest comfortably against the player’s body.
2. Water Waves
   When you throw two stones into a pond at the same time, they create ripples (waves) that spread out in circles. Where these ripples meet, they form curves that look like hyperbolas.
3. Lenses and Screens
   Devices like telescopes, microscopes, and TVs often use hyperbolic shapes in their design. These shapes help focus light properly, giving us clearer images.
4. Satellite Paths
   When scientists send satellites into space, they use the math behind hyperbolas to predict and plan the satellite's path. Hyperbolic equations help guide where and how the satellite will move.

Parts of a Hyperbola 

A hyperbola is a special type of curve formed when a flat surface (a plane) cuts through both cones of a double cone at a certain angle. This creates two open curves that face away from each other.

To better understand a hyperbola, here are the important parts:

1. Foci (plural of Focus)
   These are two fixed points inside each curve. The difference in distances from any point on the hyperbola to the two foci is always the same.
2. Directrix
   A straight line used to define the hyperbola. Each branch of the hyperbola has its own directrix. The distance of any point on the hyperbola from a focus and from the directrix follows a specific ratio.
3. Vertices
   The two closest points on the hyperbola curves. These lie on the main axis of the hyperbola and are the "turning points" of each branch.
4. Latus Rectum
   A line segment that passes through a focus and is perpendicular to the axis of the hyperbola. It helps in measuring the width of the curve near the foci.
5. Eccentricity (e)
   It tells how stretched or open the hyperbola is. For all hyperbolas, eccentricity is greater than 1.

Part	Description (Simple Explanation)
Foci	Two fixed points at (c, 0) and (−c, 0) that help define the shape of the hyperbola.
Centre	The middle point between the two foci, usually called O. It is where the axes of the hyperbola meet.
Major Axis	The longest line through the center and both ends of the curve. Its total length is 2a units.
Minor Axis	The shorter line through the center, perpendicular to the major axis. Length is 2b units.
Vertices	Points on the hyperbola where it crosses the major axis — (a, 0) and (−a, 0).
Transverse Axis	A horizontal line that passes through the foci and center. Splits the hyperbola into two parts.
Conjugate Axis	A vertical line through the center that’s at a right angle to the transverse axis.
Asymptotes	Lines the hyperbola gets closer to but never touches: y = (b/a)x and y = −(b/a)x.
Directrix	A fixed straight line that is perpendicular to the transverse axis. Used to define the curve.

Solved Examples of Equation of Hyperbola

Example 1: Find the equation of hyperbola whose vertices are (± 2, 0) and the foci are (± 3, 0).

Calculation: Vertices and foci of hyperbola are: (± 2, 0) and (± 3, 0) respectively.

As we know that for the hyperbola of the form :

The vertices and foci are given by: (± a, 0) and (± c, 0) respectively.

⇒ a = 2 and c = 3.

⇒

The vertices, co-vertices, and foci are related by the equation

As we know,

By substituting the value of in the above equation, we get:

⇒

Here we obtain the value of a= 2 and the value of b=5

Example 2: Find the equation of hyperbola whose vertices are (± 7, 0) and the eccentricity is 4/3.

Calculation: Given: The vertices of hyperbola are: (± 7, 0) and eccentricity is 4/3.

∵ The vertices of the given hyperbola are of the form (± a, 0).

So, it is a horizontal hyperbola i.e. it is of the form:

Therefore, a = 7 and

Here, eccentricity is i.e.

As we know that

⇒

The vertices, co-vertices, and foci are related by the equation

As we know that,

⇒

Hence, the required equation of hyperbola is:

Example 3: Find the equation of hyperbola whose foci are (0, ± 10) and the length of the latus rectum is 9 units.

Calculation: Given: The foci of hyperbola are (0, ± 10) and the length of the latus rectum of hyperbola is 9 units.

∵ The foci of the given hyperbola are of the form (0, ± c), it is a vertical hyperbola i.e it is of the form:

In this form of hyperbola, the center is located at the origin and foci are on the Y-axis.

⇒ c = 10 and

As we know that, the length of latus rectum of the hyperbola of the form

is given by:

⇒

⇒

The vertices, co-vertices, and foci are related by the equation

As we know that,

⇒

Substituting the values of a, b and c we obtain:

⇒

⇒

When which is not possible.

So, a = 8 and

By substituting the value of a in the equation we get

⇒

Hence, the equation of required hyperbola is

We hope that the above article on Equation of Hyperbola is helpful for your understanding and exam preparations. Stay tuned to the Testbook App for more updates on related topics from Mathematics, and various such subjects. Also, reach out to the test series available to examine your knowledge regarding several exams.

If you are checking Equation of Hyperbola article, also check the related maths articles:
Equation of Hyperbola	Parabola, Ellipse and Hyperbola
Difference Between Parabola and Hyperbola	Hyperbolic Functions
Asymptotes of Hyperbola Equation	Latus Rectum of Hyperbola

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