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