Vertex of the Hyperbola

 We will discuss about the vertex of the hyperbola along with the examples.

Definition of the vertex of the hyperbola:

The vertex is the point of intersection of the line perpendicular to the directrix which passes through the focus cuts the hyperbola.

Suppose the equation of the hyperbola be \(\frac{x^{2}}{a^{2}}\) - \(\frac{y^{2}}{b^{2}}\) = 1 then, from the above figure we observe that the line perpendicular to the directrix KZ and passing through the focus S cuts the hyperbola at A and A'.

The points A and A', where the hyperbola meets the line joining the foci S and S' are called the vertices of the hyperbola.

Therefore, the hyperbola has two vertices A and A' whose co-ordinates are (a, 0) and (- a, 0) respectively.


Solved examples to find the vertex of a hyperbola:

1. Find the coordinates of the vertices of the hyperbola 9x\(^{2}\) - 16y\(^{2}\) - 144 = 0.

Solution:

The given equation of the hyperbola is 9x\(^{2}\) - 16y\(^{2}\) - 144 = 0

Now form the above equation we get,

9x\(^{2}\) - 16y\(^{2}\) = 144

Dividing both sides by 144, we get

\(\frac{x^{2}}{16}\) - \(\frac{y^{2}}{9}\) = 1

This is the form of \(\frac{x^{2}}{a^{2}}\) - \(\frac{y^{2}}{b^{2}}\) = 1, (a\(^{2}\) > b\(^{2}\)), where a\(^{2}\) = 16 or a = 4 and b\(^{2}\) = 9 or b = 3

We know the coordinates of the vertices are (a, 0) and (-a, 0).

Therefore, the coordinates of the vertices of the hyperbola 9x\(^{2}\) - 16y\(^{2}\) - 144 = 0 are (4, 0) and (-4, 0).

 

2. Find the coordinates of the vertices of the hyperbola 9x\(^{2}\) - 25y\(^{2}\) - 225 = 0.

Solution:

The given equation of the hyperbola is 9x\(^{2}\) - 25y\(^{2}\) - 225 = 0

Now form the above equation we get,

9x\(^{2}\) - 25y\(^{2}\) = 225

Dividing both sides by 225, we get

\(\frac{x^{2}}{25}\) - \(\frac{y^{2}}{9}\) = 1

Comparing the equation \(\frac{x^{2}}{25}\) - \(\frac{y^{2}}{9}\) = 1 with the standard equation of hyperbola \(\frac{x^{2}}{a^{2}}\) - \(\frac{y^{2}}{b^{2}}\) = 1 (a\(^{2}\) > b\(^{2}\)) we get,

a\(^{2}\) = 25 or a = 5 and b\(^{2}\) = 9 or b = 3

We know the coordinates of the vertices are (a, 0) and (-a, 0).

Therefore, the coordinates of the vertices of the hyperbola 9x\(^{2}\) - 25y\(^{2}\) - 225 = 0 are (5, 0) and (-5, 0).

The Hyperbola





11 and 12 Grade Math 

From Vertex of the Hyperbola to HOME PAGE




Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.