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 

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