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# Find the equations of the hyperbola satisfying the given conditions.  vertices (± 7,0), e = 4 / 3

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

vertices (± 7,0), $e = \frac{4}{3}$

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Given, in a hyperbola

vertices (± 7,0), And

$e = \frac{4}{3}$

Here, Vertices is  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 the standard parameter (Vertices and eccentricity) with the given one, we get

$a=7$ and

$e=\frac{c}{a}=\frac{c}{7}=\frac{4}{3}$

From here,

$c=\frac{28}{3}$

Now, As we know the relation  in a hyperbola

$c^2=a^2+b^2$

$b^2=c^2-a^2$

$b^2=\left(\frac{28}{3}\right)^2-7^2$

$b^2=\left(\frac{784}{9}\right)-49$

$b^2=\left(\frac{784-441}{9}\right)=\frac{343}{9}$

Hence, The Equation of the hyperbola is ;

$\frac{x^2}{49}-\frac{9y^2}{343}=1$

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