The equation of the hyperbola is : [tex] \frac{x^{2}}{48^2} - \frac{y^{2}}{14^2} = 1 [/tex]
The center of a hyperbola is located at the origin that means at (0, 0) and one of the focus is at (-50, 0)
As both center and the focus are lying on the x-axis, so the hyperbola is a horizontal hyperbola and the standard equation of horizontal hyperbola when center is at origin: [tex] \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 [/tex]
The distance from center to focus is 'c' and here focus is at (-50,0)
So, c= 50
Now if the distance from center to the directrix line is 'd', then
[tex] d= \frac{a^{2}}{c} [/tex]
Here the directrix line is given as : x= 2304/50
Thus, [tex] \frac{a^{2}}{c} = \frac{2304}{50} [/tex]
⇒ [tex] \frac{a^{2}}{50} = \frac{2304}{50} [/tex]
⇒ a² = 2304
⇒ a = √2304 = 48
For hyperbola, b² = c² - a²
⇒ b² = 50² - 48² (By plugging c=50 and a = 48)
⇒ b² = 2500 - 2304
⇒ b² = 196
⇒ b = √196 = 14
So, the equation of the hyperbola is : [tex] \frac{x^{2}}{48^2} - \frac{y^{2}}{14^2} = 1 [/tex]
