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Find the equations of the hyperbola satisfying the given conditions. Foci (± 3 root of 5 , 0), the latus rectum is of length 8.

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

       Foci (\pm 3\sqrt5, 0), the latus rectum is of length 8.

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Given, in a hyperbola

Foci (\pm 3\sqrt5, 0), the latus rectum is of length 8.

Here,  focii are on the X-axis so, the standard equation of the Hyperbola will be ;

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

By comparing standard parameter (length of latus rectum and foci) with the given one, we get

c=3\sqrt{5} and 

\frac{2b^2}{a}=8\Rightarrow 2b^2=8a\Rightarrow b^2=4a

Now, As we know the relation  in a hyperbola 

c^2=a^2+b^2

c^2=a^2+4a

a^2+4a=(3\sqrt{5})^2

a^2+4a=45

a^2+9a-5a-45=0

(a+9)(a-5)=0

a=-9\:or\:5

Since a can never be negative,

a=5

a^2=25

b^2=4a=4(5)=20

Hence, The Equation of the hyperbola is ;

\frac{x^2}{25}-\frac{y^2}{20}=1

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