Find the equations of the hyperbola satisfying the given conditions :Vertices $\left(\pm 7,0\right)$, $e=\frac{4}{3}$

Solution: Here, Vertices of hyperbola $=\left(\pm 7,0\right)$
That means the transverse axis of the hyperbola is $X$-axis.
So, the equation will be of the type,
$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\to \left(1\right)$
Also, $a=7$
In a hyperbola, $c^2=a^2+b^2$
Putting value of $c$,
$\Rightarrow c^2=7^2+b^2$
$\Rightarrow c^2=49+b^2\to \left(2\right)$
Eccentricity of hyperbola $=\frac{4}{3}$
$\Rightarrow \frac{c}{a}=\frac{4}{3}\Rightarrow \frac{c}{7}=\frac{4}{3}\Rightarrow c=\frac{28}{3}$
Putting value of $c$ in (2),
${\left(\frac{28}{3}\right)}^2=49+b^2\Rightarrow b^2={\left(\frac{28}{3}\right)}^2-49$
$b^2=\frac{784}{9}-49=\frac{343}{9}$
Putting values of $a^2$ and $b^2$ in (1),
$\frac{x^2}{49}-\frac{9y^2}{343}=1$