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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