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Find the equations of the hyperbola satisfying the given conditions. Foci (0, +- root of 10 ), passing through (2,3)

15. Find the equations of the hyperbola satisfying the given conditions.

       Foci (0,\pm\sqrt{10}), passing through (2,3)

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Given, in a hyperbola,

 Foci (0,\pm\sqrt{10}), passing through (2,3)

Since foci of the hyperbola are in Y-axis, the equation of the hyperbola will be of the form ;

\frac{y^2}{a^2}-\frac{x^2}{b^2}=1

By comparing standard parameter (foci) with the given one, we get

c=\sqrt{10}

Now As we know, in a hyperbola 

a^2+b^2=c^2

a^2+b^2=10\:\:\:\:\:\:\:....(1)

Now As the hyperbola passes through the point (2,3)

\frac{3^2}{a^2}-\frac{2^2}{b^2}=1

9b^2-4a^2=a^2b^2\:\;\;\:\:\;\:....(2)

Solving Equation (1) and (2)

9(10-a^2)-4a^2=a^2(10-a^2)

a^4-23a^2+90=0

(a^2)^2-18a^2-5a^2+90=0

(a^2-18)(a^2-5)=0

a^2=18\:or\:5

Now, as we know that in a hyperbola c is always greater than, a we choose the value

a^2=5

b^2=10-a^2=10-5=5

Hence The Equation of the hyperbola is 

\frac{y^2}{5}-\frac{x^2}{5}=1

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