Centre of the Hyperbola

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).

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The Hyperbola