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## Equations of Hyperbolas

### Learning Outcomes

- Derive an equation for a hyperbola centered at the origin
- Write an equation for a hyperbola centered at the origin
- Solve an applied problem involving hyperbolas

In analytic geometry a **hyperbola** is a conic section formed by intersecting a right circular cone with a plane at an angle such that both halves of the cone are intersected. This intersection produces two separate unbounded curves that are mirror images of each other.

Like the ellipse, the hyperbola can also be defined as a set of points in the coordinate plane. A hyperbola is the set of all points (x,y) in a plane such that the difference of the distances between (x,y)

and the foci is a positive constant.

Notice that the definition of a hyperbola is very similar to that of an ellipse. The distinction is that the hyperbola is defined in terms of the *difference* of two distances, whereas the ellipse is defined in terms of the *sum* of two distances.

As with the ellipse, every hyperbola has two **axes of symmetry**. The **transverse axis** is a line segment that passes through the center of the hyperbola and has vertices as its endpoints. The foci lie on the line that contains the transverse axis. The **conjugate axis** is perpendicular to the transverse axis and has the co-vertices as its endpoints. The **center of a hyperbola** is the midpoint of both the transverse and conjugate axes, where they intersect. Every hyperbola also has two **asymptotes** that pass through its center. As a hyperbola recedes from the center, its branches approach these asymptotes. The **central rectangle** of the hyperbola is centered at the origin with sides that pass through each vertex and co-vertex; it is a useful tool for graphing the hyperbola and its asymptotes. To sketch the asymptotes of the hyperbola, simply sketch and extend the diagonals of the central rectangle.

In this section we will limit our discussion to hyperbolas that are positioned vertically or horizontally in the coordinate plane; the axes will either lie on or be parallel to the *x*– and *y*-axes. We will consider two cases: those that are centered at the origin, and those that are centered at a point other than the origin.

## Standard Form of the Equation of a Hyperbola Centered at the Origin

Let (−c,0) and (c,0) be the **foci** of a hyperbola centered at the origin. The hyperbola is the set of all points (x,y) such that the difference of the distances from (x,y) to the foci is constant.

If (a,0) is a vertex of the hyperbola, the distance from (−c,0) to (a,0) is a−(−c)=a+c. The distance from (c,0) to (a,0) is c−a

. The difference of the distances from the foci to the vertex is

(a+c)−(c−a)=2a

If (x,y)

is a point on the hyperbola, we can define the following variables:

d2=the distance from (−c,0) to (x,y)d1=the distance from (c,0) to (x,y)

By definition of a hyperbola, |d2−d1| is constant for any point (x,y) on the hyperbola. We know that the difference of these distances is 2a for the vertex (a,0). It follows that |d2−d1|=2a for any point on the hyperbola. The derivation of the equation of a hyperbola is based on applying the **distance formula**, but is again beyond the scope of this text. The standard form of an equation of a hyperbola centered at the origin with vertices (±a,0) and co-vertices (0±b) is

### A General Note: Standard Forms of the Equation of a Hyperbola with Center (0,0)

The standard form of the equation of a hyperbola with center (0,0) and transverse axis on the *x*-axis is

the coordinates of the co-vertices are (±b,0) the distance between the foci is 2c, where c^{2}=a^{2}+b^{2} the coordinates of the foci are (0,±c) the equations of the asymptotes are y=±^{a}/_{b}x

Note that the vertices, co-vertices, and foci are related by the equation c^{2}=a^{2}+b^{2}. When we are given the equation of a hyperbola, we can use this relationship to identify its vertices and foci.

### How To: Given the equation of a hyperbola in standard form, locate its vertices and foci.

- Determine whether the transverse axis lies on the
*x*– or*y*-axis. Notice that a^{2}

is always under the variable with the positive coefficient. So, if you set the other variable equal to zero, you can easily find the intercepts. In the case where the hyperbola is centered at the origin, the intercepts coincide with the vertices.

## Writing Equations of Hyperbolas in Standard Form

Just as with ellipses, writing the equation for a hyperbola in standard form allows us to calculate the key features: its center, vertices, co-vertices, foci, asymptotes, and the lengths and positions of the transverse and conjugate axes. Conversely, an equation for a hyperbola can be found given its key features. We begin by finding standard equations for hyperbolas centered at the origin. Then we will turn our attention to finding standard equations for hyperbolas centered at some point other than the origin.

## Hyperbolas Centered at the Origin

Reviewing the standard forms given for hyperbolas centered at (0,0), we see that the vertices, co-vertices, and foci are related by the equation c^{2}=a^{2}+b^{2}. Note that this equation can also be rewritten as b^{2}=c^{2}−a^{2}. This relationship is used to write the equation for a hyperbola when given the coordinates of its foci and vertices.

### How To: Given the vertices and foci of a hyperbola centered at (0,0)

### , write its equation in standard form.

- Determine whether the transverse axis lies on the
*x*– or*y*-axis.

## Hyperbolas Not Centered at the Origin

Like the graphs for other equations, the graph of a hyperbola can be translated. If a hyperbola is translated h units horizontally and k units vertically, the center of the **hyperbola** will be (h,k). This translation results in the standard form of the equation we saw previously, with x replaced by (x−h) and y replaced by (y−k).

### A General Note: Standard Forms of the Equation of a Hyperbola with Center (*h*, *k*)

The standard form of the equation of a hyperbola with center (h,k) and transverse axis parallel to the *x*-axis is

Like hyperbolas centered at the origin, hyperbolas centered at a point (h,k) have vertices, co-vertices, and foci that are related by the equation c^{2}=a^{2}+b^{2}. We can use this relationship along with the midpoint and distance formulas to find the standard equation of a hyperbola when the vertices and foci are given.

### How To: Given the vertices and foci of a hyperbola centered at (h,k)

### , write its equation in standard form.

- Determine whether the transverse axis is parallel to the
*x*– or*y*-axis.

### Example: Finding the Equation of a Hyperbola Centered at (*h*, *k*) Given its Foci and Vertices

What is the standard form equation of the **hyperbola** that has vertices at (0,−2) and (6,−2) and foci at (−2,−2) and (8,−2)?

**Solution**

The *y*-coordinates of the vertices and foci are the same, so the transverse axis is parallel to the *x*-axis. Thus, the equation of the hyperbola will have the form

=16

Finally, substitute the values found for h,k,a^{2}, and b^{2} into the standard form of the equation.

## Solving Applied Problems Involving Hyperbolas

As we discussed at the beginning of this section, hyperbolas have real-world applications in many fields, such as astronomy, physics, engineering, and architecture. The design efficiency of hyperbolic cooling towers is particularly interesting. Cooling towers are used to transfer waste heat to the atmosphere and are often touted for their ability to generate power efficiently. Because of their hyperbolic form, these structures are able to withstand extreme winds while requiring less material than any other forms of their size and strength. For example a 500-foot tower can be made of a reinforced concrete shell only 6 or 8 inches wide!

The first hyperbolic towers were designed in 1914 and were 35 meters high. Today, the tallest cooling towers are in France, standing a remarkable 170 meters tall. In Example 6 we will use the design layout of a cooling tower to find a hyperbolic equation that models its sides.

### Example: Solving Applied Problems Involving Hyperbolas

The design layout of a cooling tower is shown below. The tower stands 179.6 meters tall. The diameter of the top is 72 meters. At their closest, the sides of the tower are 60 meters apart.

Find the equation of the hyperbola that models the sides of the cooling tower. Assume that the center of the **hyperbola**—indicated by the intersection of dashed perpendicular lines in the figure—is the origin of the coordinate plane. Round final values to four decimal places. Show Solution

We are assuming the center of the tower is at the origin, so we can use the standard form of a horizontal hyperbola centered at

the origin:

where the branches of the hyperbola form the sides of the cooling tower. We must find the values of a^{2} and b^{2} to complete the model.

First, we find a2. Recall that the length of the transverse axis of a hyperbola is 2a. This length is represented by the distance where the sides are closest, which is given as 65.3 meters. So, 2a=60. Therefore, a=30 and a^{2}=900.

To solve for b^{2}, we need to substitute for x and y in our equation using a known point. To do this, we can use the dimensions of the tower to find some point (x,y) that lies on the hyperbola. We will use the top right corner of the tower to represent that point. Since the *y*-axis bisects the tower, our *x*-value can be represented by the radius of the top, or 36 meters. The *y*-value is represented by the distance from the origin to the top, which is given as 79.6 meters. Therefore,

**Try It**

A design for a cooling tower project is shown below. Find the equation of the hyperbola that models the sides of the cooling tower. Assume that the center of the hyperbola—indicated by the intersection of dashed perpendicular lines in the figure—is the origin of the coordinate plane. Round final values to four decimal places.

## Hyperbola

In mathematics, a hyperbola is an important conic section formed by the intersection of the double cone by a plane surface, but not necessarily at the center. A hyperbola is symmetric along the conjugate axis, and shares many similarities with the ellipse. Concepts like foci, directrix, latus rectum, eccentricity, apply to a hyperbola. A few common examples of hyperbola include the path followed by the tip of the shadow of a sundial, the scattering trajectory of sub-atomic particles, etc.

Here we shall aim at understanding the definition, formula of a hyperbola, derivation of the formula, and standard forms of hyperbola using the solved examples.

## What is Hyperbola?

A hyperbola, a type of smooth curve lying in a plane, has two pieces, called connected components or branches, that are mirror images of each other and resemble two infinite bows. A hyperbola is a set of points whose difference of distances from two foci is a constant value. This difference is taken from the distance from the farther focus and then the distance from the nearer focus. For a point P(x, y) on the hyperbola and for two foci F, F’, the locus of the hyperbola is PF – PF’ = 2a.

### Hyperbola Definition

A hyperbola, in analytic geometry, is a conic section that is formed when a plane intersects a double right circular cone at an angle such that both halves of the cone are intersected. This intersection of the plane and cone produces two separate unbounded curves that are mirror images of each other called a hyperbola.

## Parts of a Hyperbola

Let us check through a few important terms relating to the different parameters of a hyperbola.

**Foci of hyperbola:** The hyperbola has two foci and their coordinates are F(c, o), and F'(-c, 0).

**Center of Hyperbola:** The midpoint of the line joining the two foci is called the center of the hyperbola.

**Major Axis:** The length of the major axis of the hyperbola is 2a units.

**Minor Axis:** The length of the minor axis of the hyperbola is 2b units.

**Vertices:** The points where the hyperbola intersects the axis are called the vertices. The vertices of the hyperbola are (a, 0), (-a, 0).

**Latus Rectum of Hyperbola:** The latus rectum is a line drawn perpendicular to the transverse axis of the hyperbola and is passing through the foci of the hyperbola. The length of the latus rectum of the hyperbola is 2b^{2}/a.

**Transverse Axis:** The line passing through the two foci and the center of the hyperbola is called the transverse axis of the hyperbola.

**Conjugate Axis:** The line passing through the center of the hyperbola and perpendicular to the transverse axis is called the conjugate axis of the hyperbola.

**Eccentricity of Hyperbola:** (e > 1) The eccentricity is the ratio of the distance of the focus from the center of the hyperbola, and the distance of the vertex from the center of the hyperbola. The distance of the focus is ‘c’ units, and the distance of the vertex is ‘a’ units, and hence the eccentricity is e = c/a.

## Hyperbola Equation

The below equation represents the general equation of a hyperbola. Here the x-axis is the transverse axis of the hyperbola, and the y-axis is the conjugate axis of the hyperbola.

Let us understand the standard form of the hyperbola equation and its derivation in detail in the following sections.

## Standard Equation of Hyperbola

There are two standard equations of the Hyperbola. These equations are based on the transverse axis and the conjugate axis of each of the

it has the transverse axis as the y-axis and its conjugate axis is the x-axis. The below image shows the two standard forms of equations of the hyperbola.

## Derivation of Hyperbola Equation

As per the definition of the hyperbola, let us consider a point P on the hyperbola, and the difference of its distance from the two foci F, F’ is 2a.

PF’ – PF = 2a

Let the coordinates of P be (x, y) and the foci be F(c, o) and F'(-c, 0)

Therefore, the standard equation of the Hyperbola is derived.

## Hyperbola Formula

Hyperbola is an open curve that has two branches that look like mirror images of each other. For any point on any of the branches, the absolute difference between the point from foci is constant and equals to 2a, where a is the distance of the branch from the center. The Hyperbola formula helps us to find various parameters and related parts of the hyperbola such as the equation of hyperbola, the major and minor axis, eccentricity, asymptotes, vertex, foci, and semi-latus rectum.

**Equation of hyperbola formula:** (x – x0)^{2} / a^{2} – ( y – y0

)^{2} / b^{2} = 1

**Major and minor axis formula:** y = y0 is the major axis, and its length is 2a, whereas x = x0 is the minor axis, and its length is 2b

**Semi-latus rectum(p) of hyperbola formula:**

p = b^{2} / a

where,

- x0

, y0

- are the center points.
- a = semi-major axis.
- b = semi-minor axis.

**Example:** The equation of the hyperbola is given as (x – 5)^{2}/4^{2 }– (y – 2)^{2}/ 2^{2} = 1. Use the hyperbola formulas to find the length of the Major Axis and Minor Axis.

**Solution:**

Using the hyperbola formula for the length of the major and minor axis

Length of major axis = 2a, and length of minor axis = 2b

Length of major axis = 2 × 4 = 8, and Length of minor axis = 2 × 2 = 4

**Answer:** The length of the major axis is 8 units, and the length of the minor axis is 4 units.

## Graph of Hyperbola

All hyperbolas share common features, consisting of two curves, each with a vertex and a focus. The transverse axis of a hyperbola is the axis that crosses through both vertices and foci, and the conjugate axis of the hyperbola is perpendicular to it. We can observe the graphs of standard forms of hyperbola equation in the figure below. If the equation of the given hyperbola is not in standard form, then we need to complete the square to get it into standard form.

We can observe the different parts of a hyperbola in the hyperbola graphs for standard equations given below.

Here,

- If the foci lie on the x-axis, the standard form of a hyperbola can be given as,

- Coordinates of the center: (h, k).
- Coordinates of vertices: (h+a, k) and (h – a,k)
- Co-vertices correspond to b, the ” minor semi-axis length”, and coordinates of co-vertices: (h,k+b) and (h,k-b).
- Foci have coordinates (h+c,k) and (h-c,k). The value of c is given as, c
^{2 }= a^{2}+ b^{2}. - Slopes of asymptotes: y = ±(b/a)x.

## Properties of a Hyperbola

The following important properties related to different concepts help in understanding hyperbola better.

**Asymptotes: **The pair of straight lines drawn parallel to the hyperbola and assumed to touch the hyperbola at infinity. The equations of the asymptotes of the hyperbola are y = bx/a, and y = -bx/a respectively.

**Rectangular Hyperbola:** The hyperbola having the transverse axis and the conjugate axis of the same length is called the rectangular hyperbola. Here, we have 2a = 2b, or a = b. Hence the equation of the rectangular hyperbola is equal to x^{2} – y^{2} = a^{2}

**Parametric Coordinates:** The points on the hyperbola can be represented with the parametric coordinates (x, y) = (asecθ, btanθ). These parametric coordinates representing the points on the hyperbola satisfy the equation of the hyperbola.

**Auxilary Circle:** A circle drawn with the endpoints of the transverse axis of the hyperbola as its diameter is called the auxiliary circle. The equation of the auxiliary circle of the hyperbola is x^{2} + y^{2} = a^{2}.

**Direction Circle:** The locus of the point of intersection of perpendicular tangents to the hyperbola is called the director circle. The equation of the director circle of the hyperbola is x^{2} + y^{2} = a^{2} – b^{2}.

## Examples on Hyperbola

**Example 1:** The equation of the hyperbola is given as [(x – 5)^{2}/4^{2}] – [(y – 2)^{2}/ 6^{2}] = 1. Find the asymptote of this hyperbola.

**Solution:**

Using the one of the hyperbola formulas (for finding asymptotes):

y = y

– (b/a)x + (b/a)x

and y = y + (b/a)x – (b/a)xy = 2 – (6/4)x + (6/4)5 and y = 2 + (6/4)x – (6/4)5

**Answer: **Asymptotes are y = 2 – ( 3/2)x + (3/2)5, and y = 2 + 3/2)x – (3/2)5.

**Example 2:** The equation of the hyperbola is given as [(x – 5)^{2}/6^{2}] – [(y – 2)^{2}/ 4^{2}] = 1. Use the hyperbola formulas to find the length of the Major Axis and Minor Axis.

**Solution:**

Using the hyperbola formula for the length of the major and minor axis

Length of major axis = 2a, and length of minor axis = 2b

Length of major axis = 2 × 6 = 12, and Length of minor axis = 2 × 4 = 8

**Answer:** The length of the major axis is 12 units, and the length of the minor axis is 8 units.

**Example 3:** The equation of the hyperbola is given as (x – 3)^{2}/5^{2} – (y – 2)^{2}/ 4^{2} = 1. Find the asymptote of this hyperbola.

**Solution:**

Using the one of the hyperbola formulas (for finding asymptotes):

y = y

– (b/a)x + (b/a)x

and y = y – (b/a)x + (b/a)xy = 2 – (4/5)x + (4/5)5 and y = 2 + (4/5)x – (4/5)5

**Answer: **Asymptotes are y = 2 – (4/5)x + 4, and y = 2 + (4/5)x – 4.

## FAQs on Hyperbola

### What is Hyperbola in Conic Section?

A hyperbola is the locus of a point whose difference of the distances from two fixed points is a constant value. The two fixed points are called the foci of the hyperbola, and the equation of the

### How to Find the General Equation of a Hyperbola?

The equation of the hyperbola can be derived from the basic definition of a hyperbola: A hyperbola is the locus of a point whose difference of the distances from two fixed points is a constant value. Let the fixed point be P(x, y), the foci are F and F’. Then the condition is PF – PF’ = 2a. The difference is taken from the farther focus, and then the nearer focus. This on further substitutions and simplification we have the equation of the

### What is a Rectangular Hyperbola?

The hyperbola having the major axis and the minor axis of equal length is called a rectangular hyperbola. Hence we have 2a = 2b, or a = b. The equation of the rectangular hyperbola is x^{2} – y^{2} = a^{2}.

### What is the Standard Equation of Hyperbola?

There are two standard equations of the Hyperbola. These equations are given as,

### What is the Eccentricity of Hyperbola?

The eccentricity of the hyperbola is greater than 1. (e > 1). The eccentricity is the ratio of the distance of the focus from the center of the ellipse, and the distance of the vertex from the center of the ellipse. The distance of the focus is ‘c’ units, and the distance of the vertex is ‘a’ units, and hence the eccentricity is e = c/a. Also here we have c^{2} = a^{2} + b^{2}.

### What are Asymptotes of Hyperbola?

The asymptotes are the lines that are parallel to the hyperbola and are assumed to meet the hyperbola at infinity. The equation of asymptotes of the hyperbola are y = bx/a, and y = -bx/a. The

## The Hyperbola in Standard Form

A hyperbola is the set of points in a plane whose distances from two fixed points, called foci, has an absolute difference that is equal to a positive constant. In other words, if points *F*1 and *F*2 are the foci and *d* is some given positive constant then (*x*,*y*) is a point on the hyperbola if *d*=|*d*1−*d*2| as pictured below:

In addition, a hyperbola is formed by the intersection of a cone with an oblique plane that intersects the base. It consists of two separate curves, called branches. Points on the separate branches of the graph where the distance is at a minimum are called vertices. The midpoint between a hyperbola’s vertices is its center. Unlike a parabola, a hyperbola is asymptotic to certain lines drawn through the center. In this section, we will focus on graphing hyperbolas that open left and right or upward and downward.

The asymptotes are drawn dashed as they are not part of the graph; they simply indicate the end behavior of the graph. The equation of a hyperbola opening left and right in standard form follows:

Here the center is (*h*,*k*) and the vertices are (*h*±*a*,*k*). The equation of a hyperbola opening upward and downward in standard form follows:

Here the center is (*h*,*k*) and the vertices are (*h*,*k*±*b*).

The asymptotes are essential for determining the shape of any hyperbola. Given standard form, the asymptotes are lines passing through the center (*h*,*k*) with slope *m*=±* ^{b}/_{a}*. To easily sketch the asymptotes we make use of two special line segments through the center using

*a*and

*b*. Given any hyperbola, the transverse axis is the line segment formed by its vertices. The conjugate axis is the line segment through the center perpendicular to the transverse axis as pictured below:

The rectangle defined by the transverse and conjugate axes is called the fundamental rectangle. The lines through the corners of this rectangle have slopes *m*=±* ^{b}/_{a}*. These lines are the asymptotes that define the shape of the hyperbola. Therefore, given standard form, many of the properties of a hyperbola are apparent.

The graph of a hyperbola is completely determined by its center, vertices, and asymptotes.

Use these dashed lines as a guide to graph the hyperbola opening left and right passing through the vertices.

Answer:

Use these dashed lines as a guide to graph the hyperbola opening upward and downward passing through the vertices.

Answer:

**Note**: When given a hyperbola opening upward and downward, as in the previous example, it is a common error to interchange the values for the center, *h* and *k*. This is the case because the quantity involving the variable *y* usually appears first in standard form. Take care to ensure that the *y*-value of the center comes from the quantity involving the variable *y* and that the *x*-value of the center is obtained from the quantity involving the variable *x*.

As with any graph, we are interested in finding the *x*– and *y*-intercepts.

Standard form requires one side to be equal to 1. In this case, we can obtain standard form by dividing both sides by 45.

## The Hyperbola in General Form

We have seen that the graph of a hyperbola is completely determined by its center, vertices, and asymptotes; which can be read from its equation in standard form. However, the equation is not always given in standard form. The equation of a hyperbola in general form follows:

Solution:

Begin by rewriting the equation in standard form.

**Step 1:**Group the terms with the same variables and move the constant to the right side. Factor so that the leading coefficient of each grouping is 1.

## Identifying the Conic Sections

In this section, the challenge is to identify a conic section given its equation in general form. To distinguish between the conic sections, use the exponents and coefficients. If the equation is quadratic in only one variable and linear in the other, then its graph will be a parabola.

If the equation is quadratic in both variables, where the coefficients of the squared terms are the same, then its graph will be a circle.

If the equation is quadratic in both variables where the coefficients of the squared terms are different but have the same sign, then its graph will be an ellipse.

If the equation is quadratic in both variables where the coefficients of the squared terms have different signs, then its graph will be a hyperbola.

### Example 5

Identify the graph of each equation as a parabola, circle, ellipse, or hyperbola.

## Solved Problems for You

**Question 1: Find the equation of the hyperbola where foci are (0, ±12) and the length of the latus rectum is 36.**

**Answer:** The foci are (0, ±12). Hence, c = 12. Length of the latus rectum = 36 = 2b^{2}/a

∴ b^{2} = 18a

Hence, from c^{2} = a^{2} + b^{2}, we have 12^{2} = a^{2} + 18a

Or, 144 = a^{2} + 18a

i.e. a^{2} + 18a – 144 = 0

Solving it, we get a = – 24, 6

Since ‘a’ cannot be negative, we take a = 6 and so b^{2} = 36a/2 = (36 x 6)/2 = 108. Therefore, the equation of the required hyperbola is y^{2}/36 – x^{2}/108 = 1.

**Question 2: What is the focus of hyperbola?**

**Answer: **The foci of a hyperbola are the two fixed points which are situated inside each curve of a hyperbola which is useful in the curve’s formal definition.

**Question 3: What is the difference between a parabola and a hyperbola?**

**Answer: **A parabola comprises of two arms of the curve which we also refer to as branches that become parallel to each other. However, in a hyperbola, the two arms or curves do not become parallel.

**Question 4: What makes a hyperbola?**

**Answer: **An intersection of a plane perpendicular to the bases of a double cone forms a hyperbola. All the hyperbolas have two branches having a vertex and focal point. Moreover, all hyperbolas have an eccentricity value which is greater than 1.

**Question 5: What is the use of hyperbola?**

**Answer: **We see that the hyperbolic paraboloid is a 3D surface that is a hyperbola in one cross-section and a parabola in the other. Thus, we make use of hyperbolic structures in Cooling Towers of Nuclear Reactors.

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