We will discuss about the hyperbola of the ellipse along with the examples.
The centre of a conic section is a point which bisects every chord passing through it.
Definition of the Centre of the Hyperbola:
The mid-point of the line-segment joining the vertices of an hyperbola is called its centre.
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 C is the mid-point of the line-segment AA', where A and A' are the two vertices. In case of the hyperbola \(\frac{x^{2}}{a^{2}}\) - \(\frac{y^{2}}{b^{2}}\) = 1, every chord is bisected at C (0, 0).
Therefore, C is the centre of the hyperbola and its co-ordinates are (0, 0).
Solved examples to find the centre of an hyperbola:
1. Find the co-ordinates of the centre of the hyperbola 3x\(^{2}\) - 2y\(^{2}\) - 6 = 0.
Solution:
The given equation of the hyperbola is 3x\(^{2}\) - 2y\(^{2}\) - 6 = 0.
Now form the above equation we get,
3x\(^{2}\) - 2y\(^{2}\) - 6 = 0
⇒ 3x\(^{2}\) - 2y\(^{2}\) = 6
Now dividing both sides by 6, we get
\(\frac{x^{2}}{2}\) - \(\frac{y^{2}}{3}\) = 1 ………….. (i)
This equation is of the form \(\frac{x^{2}}{a^{2}}\) - \(\frac{y^{2}}{b^{2}}\) = 1 (a\(^{2}\) > b\(^{2}\)).
Clearly, the centre of the hyperbola (1) is at the origin.
Therefore, the co-ordinates of the centre of the hyperbola 3x\(^{2}\) - 2y\(^{2}\) - 6 = 0 is (0, 0)
2. Find the co-ordinates of the centre the hyperbola 5x\(^{2}\) - 9y\(^{2}\) - 10x + 90y + 185 = 0.
Solution:
The given equation of the hyperbola is 5x\(^{2}\) - 9y\(^{2}\) - 10x - 90y - 265 = 0.
Now form the above equation we get,
5x\(^{2}\) - 9y\(^{2}\) - 10x - 90y - 265 = 0
⇒ 5x\(^{2}\) - 10x + 5 - 9y\(^{2}\) - 90y - 225 - 265 - 5 + 225 = 0
⇒ 5(x\(^{2}\) - 2x + 1) - 9(y\(^{2}\) + 10y + 25) = 45
⇒ \(\frac{(x - 1)^{2}}{9}\) - \(\frac{(y + 5)^{2}}{5}\) = 1
We
know that the equation of the hyperbola having centre at (α, β) and major and minor axes parallel to x and y-axes
respectively is, \(\frac{(x - α)^{2}}{a^{2}}\) - \(\frac{(y - β)^{2}}{b^{2}}\) = 1.
Now, comparing equation \(\frac{(x - 1)^{2}}{9}\) - \(\frac{(y + 5)^{2}}{5}\) = 1 with equation \(\frac{(x - α)^{2}}{a^{2}}\) - \(\frac{(y - β)^{2}}{b^{2}}\) = 1 we get,
α = 1, β = - 5, a\(^{2}\) = 9 ⇒ a = 3 and b\(^{2}\) = 5 ⇒ b = √5.
Therefore, the co-ordinates of its centre are (α, β) i.e., (1, - 5).
● The Hyperbola
11 and 12 Grade Math
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