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Find the equations of the hyperbola satisfying the given conditions. Foci (± 4, 0), the latus rectum is of length 12

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

      Foci (± 4, 0), the latus rectum is of length 12

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Given, in a hyperbola

Foci (± 4, 0), the latus rectum is of length 12

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=4 and 

\frac{2b^2}{a}=12\Rightarrow 2b^2=12a\Rightarrow b^2=6a

Now, As we know the relation  in a hyperbola 

c^2=a^2+b^2

c^2=a^2+6a

a^2+6a=4^2

a^2+6a=16

a^2+8a-2a-16=0

(a+8)(a-2)=0

a=-8\:or\:2

Since a can never be negative,

a=2

a^2=4

b^2=6a=6(2)=12

Hence, The Equation of the hyperbola is ;

\frac{x^2}{4}-\frac{y^2}{12}=1

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