Find the equation of the hyperbola if the center is $\left(1, -2\right)$; length of transverse axis is 8; $e = \dfrac{5}{4}$ and the transverse axis parallel to the X axis.
Given: Center $C \left(h, k\right) = \left(1, -2\right)$; $\;$ $e = \dfrac{5}{4}$
The transverse axis is parallel to the X axis.
$\therefore$ $\;$ Let the equation of the required hyperbola be: $\;$ $\dfrac{\left(x - h\right)^2}{a^2} - \dfrac{\left(y - k\right)^2}{b^2} = 1$
Length of transverse axis $= 2a = 8$ $\implies$ $a = 4$
Now, $b^2 = a^2 \left(e^2 - 1\right)$
i.e. $\;$ $b^2 = 16 \times \left(\dfrac{25}{16} - 1\right) = 16 \times \dfrac{9}{16} = 9$
$\therefore$ $\;$ Equation of required hyperbola is: $\;$ $\dfrac{\left(x -1\right)^2}{16} - \dfrac{\left(y + 2\right)^2}{9} = 1$