Find the equations of the hyperbola satisfying the given conditions Foci (0, ±$$\sqrt{10}$$), passing through (2, 3)

Asked by Pragya Singh | 1 year ago |  108

Solution :-

Foci (0, ±$$\sqrt{10}$$) and passing through (2, 3)

Here, the foci are on y-axis.

The equation of the hyperbola is of the form

$$\dfrac{x^2}{a^2}$$ +$$\dfrac{y^2}{b^2}$$ = 1

Since, the foci are (±$$\sqrt{10}$$, 0), so, c =$$\sqrt{10}$$

It is known that, a2 + b2 = c2

b2 = 10 – a2 ………….. (1)

It is given that the hyperbola passes through point (2, 3)

So, $$\dfrac{9}{a^2}$$ – $$\dfrac{4}{b^2}$$ = 1 … (2)

From equations (1) and (2), we get,

$$\dfrac{9}{a^2}-\dfrac{10}{a^2}=1$$

9(10 – a2) – 4a2 = a2(10 –a2)

90 – 9a2 – 4a2 = 10a2 – a4

a4 – 23a2 + 90 = 0

a4 – 18a2 – 5a+ 90 = 0

a2(a2 -18) -5(a2 -18) = 0

(a2 – 18) (a2 -5) = 0

a2 = 18 or 5

In hyperbola, c > a i.e., c> a2

So, a2 = 5

b2 = 10 – a2

= 10 – 5

= 5

The equation of the hyperbola is

$$\dfrac{y^2}{5}- \dfrac{x^2}{5}=1$$

Answered by Abhisek | 1 year ago

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