Asymptotes of Hyperbola: Equation by Factoring & Solving for y

Asymptotes of hyperbola are the lines that pass through the center of the hyperbola. The hyperbola gets closer and closer to the asymptotes, but never touches them. Every hyperbola has two asymptotes. Hyperbola is defined as an open curve having two branches that are mirror images of each other. It is two curves that are like infinite bows whereas an asymptote is a straight line that constantly approaches a given curve but does not touch the curve.

Asymptotes: A straight line that approaches the curve on a graph but never meets the curve. That straight line is called Asymptote. 

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Hyperbola: A hyperbola is a conic section created by intersecting a right circular cone with a plane at an angle such that both halves of the cone are crossed in analytic geometry. This intersection yields two unbounded curves that are mirror reflections of one another. In other words, A hyperbola is defined as the locus of all points in a plane whose absolute difference of distances from two fixed points on the plane remains constant.

Asymptote is a straight line that approaches the curve on a graph but never meets the curve. The branches of the hyperbola approach the asymptotes but never touch them. Asymptotes of a hyperbola are the lines that pass through the center of the hyperbola. The hyperbola gets closer and closer to the asymptotes, but can never reach them. Every hyperbola has two asymptotes, which intersect the centre of the hyperbolas. The equation of the asymptotes can have four possible variations depending on the location of the centre and the orientation of the hyperbola.

Equation of Asymptotes of Hyperbola Centred at Origin

Hyperbolas centred on the origin can be oriented horizontally and vertically. Depending on this, the equation of the hyperbola is also different.

Consider the hyperbola that is centered at the origin and horizontally oriented, then the equation of the hyperbola is

Here, ‘a’ is the length of the distance from the center to a vertex, and ‘b’ is the length of the distance from the center to the co-vertex. This equation applies when the traverse axis is on the x-axis. In this case, the equation of the asymptotes of the hyperbola is given by

.

Now, consider the hyperbola that is centered at the origin and vertically oriented, then the equation of the hyperbola is

This equation applies when the traverse axis is on the y-axis. In this case, equation of the asymptotes of hyperbola is given by

.

Equation of Asymptotes of Hyperbola Centred Outside the Origin

Considering the hyperbola that is centered outside the origin, we can apply translations to obtain a new equation of the hyperbola. If the hyperbola is oriented horizontally, its equation is

Here, ‘h’ is the x coordinate of the center and ‘k’ is the ycoordinate of the center. This equation applies when the traverse axis is parallel to the x-axis. In this case, the equation of the asymptotes of hyperbola is given by

.

Now, consider the hyperbola that is centered outside the origin and vertically oriented, then the equation of the hyperbola is

 

This equation applies when the traverse axis is parallel to the y-axis. In this case, the equation of the asymptotes of hyperbola is given by

.

Angle between Asymptotes of Hyperbola

The angle between the two asymptotes of a hyperbola is , where ‘’ is the length of the distance from the center to a vertex and ‘’ is the length of the distance from the center to the co-vertex.

Proof: We know the equation of the asymptotes of the hyperbola are

Now, slope of 1st asymptote is given as:

Similarly, slope of 2nd asymptote is given as:

The angle between the asymptotes is given by

.

How to find Asymptotes of Hyperbola?

Asymptotes of a hyperbola are the lines that pass through the center of the hyperbola. The hyperbola gets closer and closer to the asymptotes, but can never reach them. There are two different methods to find the asymptotes of hyperbola which are given below:

1.By Factoring

2.By Solving for y

Find Asymptotes of Hyperbola By Factoring

Step 1: Write down the equation of the hyperbola in its standard form.

Step 2: Set the equation equal to zero instead of one.

Step 3: Factor the new equation (factor the left-hand side of the equation into two products).

Step 4: Separate the factors and solve for y.

Step 5: Try the same process with a harder equation.

For example, find the asymptotes of a hyperbola: .

Step 1:

Step 2:

Step 3:

Step 4: , and

Therefore, the asymptotes of the given hyperbola are , and .

Find Asymptotes of Hyperbola By Solving for y

Step 1: Write down the hyperbola equation with the y2 term on the left side.

Step 2: Take the square root of each side. Take the square root, but don’t try to simplify the right-hand side yet. Remember, when you take the square root, there are two possible solutions, a positive and a negative.

Step 3: Adjust the equation for large values of x.

Step 4: Solve for y to find the two asymptote equations.

Note: This method is useful if you have an equation that’s in general quadratic form. Even if it’s in standard form for hyperbolas, this approach can give you some insight into the nature of asymptotes.

For example, find the asymptotes of a hyperbola: .

Step 1:

Step 2:

Step 3: If , then

Step 4:

Now , and

.

Therefore, the asymptotes of the given hyperbola are , and .

Properties of Asymptotes of Hyperbola

Some of the important properties of asymptotes of the hyperbola are listed below:

• The product of the perpendicular from any point on the hyperbola to its asymptotes is .
• The equation of a hyperbola and its asymptotes always differ by a constant.
• Any straight line parallel to an asymptote of a hyperbola intersects the hyperbola at only one point.
• The asymptotes always pass through the center of the hyperbola.
• The angle between the two asymptotes of a hyperbola is .

Solved Examples of Asymptotes of Hyperbola

Example 1: If the angle between the asymptotes of a hyperbola is \(30^\circ/) then find its eccentricity.

Solution: If the angle between the asymptotes of a hyperbola is \(30^\circ/), we can use the relationship between the eccentricity \(e/) and the angle \(\theta/) given by \(\tan\theta = \frac{1}{e}/) to find the eccentricity.

Given the angle between the asymptotes as \(30^\circ/).
We know, \(\tan\theta = \frac{1}{e}/) 

Substitute \(\theta = 30^\circ/).
\(\tan(30^\circ) = \frac{1}{e}/).
Using the known value of \(\tan(30^\circ) = \frac{1}{\sqrt{3}}/), we can solve for \(e/):
\(\frac{1}{\sqrt{3}} = \frac{1}{e}/).
Simplifying, we find:
\(e = \sqrt{3}/).

Therefore, the eccentricity of the hyperbola when the angle between the asymptotes is \(30^\circ/) is \(\sqrt{3}/).
Example 2: If the angle between the asymptotes of a hyperbola is \(60^\circ/), find the eccentricity of the conjugate hyperbola.

Solution: If the angle between the asymptotes of a hyperbola is \(60^\circ/), we can find the eccentricity of the conjugate hyperbola using the relationship \(\tan\theta = \sqrt{e’^2 - 1}/), where \(\theta/) is the angle and \(e’/) is the eccentricity of the conjugate hyperbola.

Given: The angle \(\theta/) between the asymptotes is \(60^\circ/).

The relationship between the angle \(\theta/) and eccentricity of the conjugate hyperbola \(e’/) is \(\tan\theta = \sqrt{e’^2 - 1}/).

Substituting \(\theta = 60^\circ/) into the formula, we have:
\(\tan(60^\circ) = \sqrt{e’^2 - 1}/).

We know, \(\tan(60^\circ) = \sqrt{3}/),
Thus, \(\sqrt{3} = \sqrt{e’^2 - 1}/).
Squaring both sides of the equation, we get:
\(3 = e’^2 - 1/).
Rearranging the equation, we find:
\(e’^2 = 4/).
Taking the square root of both sides, we get:
\(e’ = 2/).

Therefore, the eccentricity of the conjugate hyperbola, given that the angle between the asymptotes is \(60^\circ/), is \(2/).

Example 3: What are the asymptotes of the hyperbola xy=4x+3y?

Solution: Rewrite the equation in standard form
The given equation is /(xy = 4x + 3y/). To factorize it, we rearrange the terms:
/(xy - 4x - 3y = 0/).

To factorize the equation, we look for common factors:
/(x(y - 4) - 3(y - 4) = 0/).
/((x - 3)(y - 4) = 0/).
We know that the asymptote condition so we set each factor equal to zero

/(x - 3 = 0/) and /(y - 4 = 0/).

Solving the equations we get,
Solving /(x - 3 = 0/), we find /(x = 3/).
Solving /(y - 4 = 0/), we find /(y = 4/).

Thus our asymptotes are x=3 and y=4

Example 4: What is the equation of the asymptotes of the hyperbola .

Solution: We can see that the equation has the form . From this we have that the hyperbola is centered at the origin and is oriented horizontally. Therefore, the equation of its asymptotes is:-

From the given equation, we have

, and

Put the values of and in equation of asymptotes, we get

Therefore, the equation of asymptotes for the given hyperbola is .

Example 5: If the hyperbola has the equation , what are its asymptotes?

Solution: We can see that the equation has the form . From this, we have that the hyperbola is centered at the origin and is oriented vertically. Therefore, the equation of its asymptotes is:-

From the given equation, we have

, and

Put the values of and in equation of asymptotes, we get

Therefore, the equation of asymptotes for the given hyperbola is .

If you are checking Asymptotes of Hyperbola article, also check the related maths articles:
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Difference between average and mean	Fourier series
Conjugate	Combinatorics

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