Q

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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