Hyperbola is an open curve with two branches that are mirror images of each other. It is two curves that look like infinite bows. In this lesson, we will explore the hyperbola equation, focii, eccentricity, directrix, latus rectum and characteristics of this type of curve.

Hyperbola is a type of conic section, which is a curve created by the intersection of a plane and a double-cone. It has two branches that extend outward from a central point and is shaped like a sideways “S”.

A hyperbola is a locus of points such that the distance to each focus is always greater than one. In other words, the locus of a point moving in a plane is determined by the ratio of its distance from a fixed point (focus) to its distance from a fixed line (directrix), and that ratio is always greater than 1.

The point does not lie on the line of the directrix.

(e > 1) $\Rightarrow$ (PS/PM) = (eccentricity)

Standard Equation of a Hyperbola

The standard equation of a hyperbola is simplified when the center of the hyperbola is at the origin and the foci are on either the x-axis or the y-axis. The equation is given as:

[(x^2 / a^2) - (y^2 / b^2)] = 1

b2 = a2(e2 - 1)

Important Terms and Formulas Related to Hyperbolas

Some of the most important terms related to hyperbola that need to be thoroughly understood in order to gain confidence in this concept are:

Eccentricity (e): $$e^2 = 1 + \frac{b^2}{a^2} = 1 + \left(\frac{\text{conjugate axis}}{\text{transverse axis}}\right)^2$$

Focii: S = (ae, 0) & S’ = (-ae, 0)

Directrix: x = $\frac{a}{e}$, x = $\frac{-a}{e}$

Transverse Axis:

The live segment A’A of length 2a in which the focii S’ and S both lie is called the transverse axis of the hyperbola.

Conjugate axis:

The line segment B’B of length $2b$ between the points B’ $(0, -b)$ and B $(0, b)$ is called the conjugate axis of the hyperbola.

Principal Axes:

The transverse axis and conjugate axis.

Vertices:

A = $(a, 0)$ and $A’ = (-a, 0)$

Focal Chord:

A chord that passes through a focus is called a focal chord.

Double ordinate:

A chord perpendicular to the transverse axis is referred to as a double ordinate.

Latus Rectum:

The line segment perpendicular to the transverse axis that passes through the focus of the parabola is referred to as the latus rectum.

Its length = $(2b^2/a) = [(conjugate)^2/transverse] = 2a(e^2 - 1)$

The difference in focal distances remains the same.

| |PS - PS’| | = |2a|

The length of the latus rectum is equal to 2e multiplied by the distance of the focus from the corresponding directrix.

Endpoints of L.R.: (± $\sqrt{ae}, \pm \frac{b}{2a}$)

Centre:

The point which bisects every chord of the conic, drawn through it, is called the centre of the conic.

The centre of the equation [(x2 / a2) - (y2 / b2)] = 1 is (0, 0).

Important:

You will notice that the results for ellipse are also applicable for a hyperbola. You need to replace b2  by (-b2)

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Practice Problems Involving Hyperbolas

Example 1:

This sentence needs to be rewritten.

This sentence has been rewritten.

The equation of the hyperbola with directrix 2x + y = 1, focus (1, 2) and eccentricity √3 is (x - 1)^2/3 - (y - 2)^2/9 = 1.

Given:

This is a heading

Solution:

This is a heading

Let P(x, y) be any point on the hyperbola.

Draw a line perpendicular to the directrix from point P.

Then, by definition, ePM = SP.

(SP)² = e² (PM)²

$(x-1)^2 + (y-2)^2 = 3\left(\frac{2x+y-1}{\sqrt{4+1}}\right)^2$

5(x^2 + y^2 - 2x - 4y + 5)

3(4x^2 + y^2 + 1 + 4xy - 2y - 4x)

7x^2 - 2y^2 + 12xy - 2x + 14y - 22 = 0

Which is the required hyperbola?

Example 2:

Find the eccentricity of the hyperbola whose latus rectum is half of its transverse axis.

Given:

This is a heading

Solution:

This is a heading

The equation of a hyperbola can be expressed as:

(x^2/a^2) - (y^2/b^2) = 1

Then transverse axis = 2a and latus rectum = (2b^2 / a)

(2b^2)/a = (1/2) * 2a

2b² = a² (since, b² = a² (e² − 1))

2a² (e² - 1) = a²

2e2 - 2 = 400 - 2 = 398

e2 = 1.5

e = $\sqrt{\frac{3}{2}}$

Hence, the required eccentricity is $\sqrt{\frac{3}{2}}$

Conjugate Hyperbola is a type of hyperbola that is composed of two separate branches which are mirror images of each other.

Two hyperbolas such that the transverse axis of one hyperbola is the conjugate axis of the other and the conjugate axis of one hyperbola is the transverse axis of the other are called conjugate hyperbolas of each other.

The equations (x² / a²) - (y² /b²) = 1 and (−x² / a²) + (y² / b²) = 1 represent conjugate hyperbolas of each other.

(y2 - b2) / (x2 - a2) = 1

a^2 = b^2 * (e^2 - 1)

Vertices: (0, $\pm$b) and L.R. = $\frac{2a^2}{b}$

Important Conclusions Regarding Conjugate Hyperbolas

If the eccentricities of the hyperbola and its conjugate are given,

1/e<sub>1</sub><sup>2</sup> + 1/e<sub>2</sub><sup>2</sup> = 1

The foci of a hyperbola and its conjugate are concyclic and form the vertices of a square, which is a four-sided polygon with four equal sides and four equal angles.

Two hyperbolas are similar if they have the same eccentricities (c).

Two similar hyperbolas are equal if they have the same latus rectum.

Example 3:

Find the following properties of the hyperbola 16x2 − 9y2 = −144:

• Lengths of transverse and conjugate axis
• Eccentricity
• Co-ordinates of foci
• Vertices
• Length of the latus-rectum
• Equations of the directrices

Given:

This is a statement

Solution:

This is a statement

(x^2 / 9) - (y^2 / 16) = -1

(x^2/a^2) - (y^2/b^2) = -1

a² = 9, b² = 16

a = 3, b = 4

Length of transverse axis: The length of the transverse axis (2b) is equal to 8.

Length of conjugate axis: The length of conjugate axis = 2a = 6

Eccentricity: Definition: the quality of being different from the norm or unusual in behaviour or appearance.

Foci: The co-ordinates of the foci are (0, $\pm$ 5) i.e., (0, $\pm$ 5)

Vertices: The coordinates of the vertices are (0, 4) and (0, -4).

Lengths of latus-rectum: The length of latus-rectum = $$\frac{2a^2}{b}$$ = $$\frac{2(3^2)}{4}$$ = $$\frac{9}{2}$$

The Equation of Directrices: The equation of directrices is x^2/a^2 - y^2/b^2 = 1.

y = ± $\frac{4}{\frac{5}{4}}$ ⇒ y = ± $\frac{16}{5}$

Auxiliary Circles of the Hyperbola

A circle with centre C and a diameter as its transverse axis is known as the auxiliary circle of a hyperbola. The equation of the auxiliary circle of a hyperbola is given as:

The equation of the auxiliary circle is: (x^2) + (y^2) = (a^2)

It can be seen from the figure that P and Q are referred to as the “corresponding points” of the hyperbola and the auxiliary circle.

Parametric Representation:

x = asec $\theta$ and y = btan $\theta$

If $(a \sec \theta, b \tan \theta)$ is on the hyperbola, then $(a \cos \theta, a \sin \theta)$ lies on the auxiliary circle. The equation of the chord joining two points $P(\alpha)$ and $Q(\phi)$ is given by:

![Hyperbola Equation]()

The Position of Point P

S1 is positive, zero, or negative depending on whether (x1, y1) lies inside, on, or outside the equation x12/a2 + y2/b2 = 1.

The point (5, -4) is outside of the hyperbola 9x2 - y2 = 1.

Given Text:

What is the meaning of life?

Solution:

The meaning of life is subjective and can vary from person to person.

Since $9^2 - (-4)^2 - 1 = 225 - 16 - 1 = 208 > 0,

The point (5, -4) lies inside the hyperbola 9x^2 - y^2 = 1.

Rectangular Hyperbola

A rectangular hyperbola has its hyperbola axes (or asymptotes) perpendicular, and its eccentricity is equal to √2. An example of a rectangular hyperbola is one with its conjugate axis equal to its transverse axis, i.e. when a = b.

x^2/a^2 - y^2/b^2

x^2/a^2 - y^2/a^2 = 1

x2 - y2 = a2

Eccentricity of a Rectangular Hyperbola

Also, (xy = c)

x = 1/y, y = 1/x

Tangent of a Rectangular Hyperbola

A tangent of a rectangular hyperbola is a line that touches a point on the rectangular hyperbola’s curve. The equation and slope form of a rectangular hyperbola’s tangent can be expressed as:

Tangent Equation

The hyperbola x^2/a^2 - y^2/b^2 = 1 will be tangent to the line y = mx + c if c^2 = a^2/m^2 - b^2.

Slope-Intercept Form of a Tangent

y = mx $\pm$ $\sqrt{a^2m^2 - b^2}$

Secant

Secant will intersect an ellipse at two distinct points

c^2 > a^2 m^2 - b^2

Neither Secant Nor Tangent

For a line to be neither secant nor tangent, a quadratic equation will yield an imaginary solution.

⇒ c² < a²m² - b²

Equation of tangent to hyperbola $x^2/a^2 - y^2/b^2 = 1$ at point $(x_1, y_1)$ is

(xx1)/a2 = (yy1)/b2 = 1

Parametric form of tangent:

$$x = x_0 + r \cos \theta$$

$$y = y_0 + r \sin \theta$$

$\frac{x\sec{\theta}}{a} - \frac{y\tan{\theta}}{b} = 1$

Point of Contact and Examples of Tangents

Contrast:

y = mx + c

(xx1/a2) - (yy1/b2) = 1 - mx + y - c

x1 = (-a2c)/m;

y1 = \frac{-b}{c}

(x1, y1) = \left[\frac{-a^2m}{c}, \frac{-b^2}{c}\right]

Solved Examples on Hyperbola

Example 5: Find the equation of the tangent to the hyperbola $\frac{x^2}{9}-y^2=1$ whose slope is 5

Given:

Welcome to our website!

Solution:

Welcome to our website! :smile:

Slope of tangent m = 5, a^2 = 9, b^2 = 1

The slope-intercept form of the equation of a tangent is:

$$y = mx + b$$

y = mx $\pm$ $\sqrt{a^2m^2 - b^2}$

y = 5x $\pm$ $\sqrt{9.52 - 1}$

y = 5x $\pm$ 4$\sqrt{14}$

The equation for a director circle of an ellipse is x²/a² + y²/b² = 1, where x² + y² = a² + b².

Normal: I like to go to the beach

Rewrite: I enjoy going to the beach.

Equation of the normal at $(x_1, y_1)$ to the ellipse $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ is $ \frac{x_1}{a^2} - \frac{y_1}{b^2} = 1$

a2x/x1 + b2y/y1 = (a2 + b2)

Example 6: Find the normal at the point (6, 3) on the hyperbola x^2/18 - y^2/9 = 1

Given:

This is a header

Solution:

This is a header

The equation of the normal at point ($x_1$, $y_1$) is $a^2 = 18$, $b^2 = 9$

a2x/x1 + b2y/y1 = (a2 + b2)(x1y1)/(x1y1)

Equation of Normal at point (6, 3) is:
$$y - 3 = \frac{1}{2} (x - 6)$$

18x/6 + 9y/3 = 18 + 9

(18x + 54y)/6 = 27

x + y = 9

Example 7: Find the equation of the chord of contact of the point (2, 3) to the hyperbola $\frac{x^2}{16} - \frac{y^2}{9} = 1$

Given:

This is a statement

Solution:

This is a statement

The equation of the chord of contact is T = 0

i.e. $\frac{xx1}{a2} - \frac{yy1}{b2} - 1 = 0$

2x/16 - 3y/9 = 1

x/8 - y/3 = 1

#Equation of a Chord when Mid-Point is Given

T = $\frac{x_1}{a^2} - \frac{y_1}{b^2} - 1 = \frac{x_2}{a^2} - \frac{y_2}{b^2} - 1$

Example 9: Find the equation of the chord of the hyperbola $\frac{x^2}{9} - \frac{y^2}{4} = 1$ whose midpoint is $(5, 1)$.

Given:

I like to play soccer

Solution:

I enjoy playing soccer

We know the equation of the chord of the hyperbola whose midpoint is $(x_1, y_1)$.

T = $\frac{x_1^2}{a^2} - \frac{y_1^2}{b^2} - 1 = \frac{x_2^2}{a^2} - \frac{y_2^2}{b^2} - 1$

The midpoint is (5, 1).

=>5x/9 - y/4 - 1 = 25/9 - ¼ - 1

=> 5x/9 - y/4 = 91/36

Important JEE Main Questions on Ellipse and Hyperbola

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Frequently Asked Questions

A Hyperbola is a type of curve defined by a mathematical equation, where the difference between the distances of two points on the curve from two fixed points is constant.

Hyperbola is the locus of a point moving in a plane where the ratio of its distance from a fixed point to its distance from a fixed line is a constant greater than 1.

The eccentricity of a hyperbola is always greater than 1.

The eccentricity of a Hyperbola is greater than 1.

The formula for eccentricity of a Hyperbola is e = √(1 + (b^2/a^2)), where a and b are the lengths of the semi-major and semi-minor axes respectively.

The eccentricity is given by:

$$e = \sqrt{1 + \frac{b^2}{a^2}}$$

The standard equation of a Hyperbola is: $$\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1$$

The standard equation of a hyperbola is (x²/a²) - (y²/b²) = 1.

A transverse axis of a hyperbola is a line segment that is perpendicular to the hyperbola’s two branches and passes through the center of the hyperbola.

The transverse axis of a hyperbola is a line passing through the center and two foci of the hyperbola. For the hyperbola (x2/a2) - (y2/b2) = 1, the transverse axis is along the x-axis, and its length is given by 2a.

An asymptote of a hyperbola is a line that the graph of the hyperbola gets infinitely close to, but never touches.

The lines that are parallel to the hyperbola and are assumed to intersect the hyperbola at infinity are referred to as the asymptotes of the hyperbola.

The foci of a hyperbola are two points located inside the hyperbola such that the sum of the distances from each point to every point on the hyperbola is constant.

The foci of a hyperbola are two points on the axis of the hyperbola that are equidistant from the center. For the hyperbola (x²/a²) - (y²/b²) = 1, the foci are given by (ae, 0) and (-ae, 0).

Conjugate axes of a hyperbola are the two lines that intersect at the center of the hyperbola and are perpendicular to each other.

The conjugate axis of a hyperbola is the line passing through the center of the hyperbola and perpendicular to the transverse axis.