Information : The given hyperbola is a horizontal hylerbola with its centre (3 , -5) and one of its focus at (9 , -5) and vertex at (7 , -5) and as we can see that the focus and vertex have same y - coordinates, it must have its Transverse axis on line y = - 5.
Now,
it's vertex is given, I.e (7 , -5)
so, length of semi transverse axis will be equal to distance of vertex from centre, i.e
Now, it's focus can be represented as ;
![\qquad \sf \dashrightarrow \: (3 + ae, - 5 )](https://tex.z-dn.net/?f=%5Cqquad%20%5Csf%20%20%5Cdashrightarrow%20%5C%3A%20%283%20%2B%20ae%2C%20%20-%205%20%29)
so,
and we know, a = 4
![\qquad \sf \dashrightarrow \: 4e + 3 = 9](https://tex.z-dn.net/?f=%5Cqquad%20%5Csf%20%20%5Cdashrightarrow%20%5C%3A%204e%20%2B%203%20%3D%209)
![\qquad \sf \dashrightarrow \: 4e = 6](https://tex.z-dn.net/?f=%5Cqquad%20%5Csf%20%20%5Cdashrightarrow%20%5C%3A%204e%20%3D%206)
![\qquad \sf \dashrightarrow \: e = \cfrac{3}{2}](https://tex.z-dn.net/?f=%5Cqquad%20%5Csf%20%20%5Cdashrightarrow%20%5C%3A%20e%20%3D%20%20%5Ccfrac%7B3%7D%7B2%7D%20)
Now, let's find the measure of semi - conjugate axis (b)
![\qquad \sf \dashrightarrow \: {b}^{2} = {a}^{2} ( {e}^{2} - 1)](https://tex.z-dn.net/?f=%5Cqquad%20%5Csf%20%20%5Cdashrightarrow%20%5C%3A%20%20%7Bb%7D%5E%7B2%7D%20%20%3D%20%20%7Ba%7D%5E%7B2%7D%20%28%20%7Be%7D%5E%7B2%7D%20%20-%201%29)
![\qquad \sf \dashrightarrow \: {b}^{2} = 16( \frac{9}{4} - 1)](https://tex.z-dn.net/?f=%5Cqquad%20%5Csf%20%20%5Cdashrightarrow%20%5C%3A%20%20%7Bb%7D%5E%7B2%7D%20%20%3D%2016%28%20%5Cfrac%7B9%7D%7B4%7D%20%20-%201%29)
![\qquad \sf \dashrightarrow \: {b}^{2} = 16( \frac{9 - 4}{4} )](https://tex.z-dn.net/?f=%5Cqquad%20%5Csf%20%20%5Cdashrightarrow%20%5C%3A%20%20%7Bb%7D%5E%7B2%7D%20%20%3D%2016%28%20%5Cfrac%7B9%20-%204%7D%7B4%7D%20%20%29)
![\qquad \sf \dashrightarrow \: {b}^{2} = 16( \frac{5}{4} )](https://tex.z-dn.net/?f=%5Cqquad%20%5Csf%20%20%5Cdashrightarrow%20%5C%3A%20%20%7Bb%7D%5E%7B2%7D%20%20%3D%2016%28%20%5Cfrac%7B5%7D%7B4%7D%20%20%29)
![\qquad \sf \dashrightarrow \: {b}^{2} = 20](https://tex.z-dn.net/?f=%5Cqquad%20%5Csf%20%20%5Cdashrightarrow%20%5C%3A%20%20%7Bb%7D%5E%7B2%7D%20%20%3D%2020)
![\qquad \sf \dashrightarrow \: b = \sqrt{20}](https://tex.z-dn.net/?f=%5Cqquad%20%5Csf%20%20%5Cdashrightarrow%20%5C%3A%20b%20%3D%20%20%20%5Csqrt%7B20%7D%20)
So, it's time to write the equation of hyperbola, as we already have the values of a and b ~
![\qquad \sf \dashrightarrow \: \cfrac{ {(x - h)}^{2} }{ {a}^{2} } - \cfrac{( {y - k)}^{2} }{ {b}^{2} } = 1](https://tex.z-dn.net/?f=%5Cqquad%20%5Csf%20%20%5Cdashrightarrow%20%5C%3A%20%20%5Ccfrac%7B%20%7B%28x%20-%20h%29%7D%5E%7B2%7D%20%7D%7B%20%7Ba%7D%5E%7B2%7D%20%7D%20%20-%20%20%5Ccfrac%7B%28%20%7By%20%20-%20k%29%7D%5E%7B2%7D%20%7D%7B%20%7Bb%7D%5E%7B2%7D%20%7D%20%20%3D%201)
[ plug in the values, and h = x - coordinate of centre, and k = y - coordinate of centre ]
![\qquad \sf \dashrightarrow \: \cfrac{ ({x-3)}^{2} }{ {16}^{} } - \dfrac{ {(y+5)}^{2} }{ { {20} }^{} } = 1](https://tex.z-dn.net/?f=%5Cqquad%20%5Csf%20%20%5Cdashrightarrow%20%5C%3A%20%20%5Ccfrac%7B%20%28%7Bx-3%29%7D%5E%7B2%7D%20%7D%7B%20%7B16%7D%5E%7B%7D%20%7D%20%20-%20%20%5Cdfrac%7B%20%7B%28y%2B5%29%7D%5E%7B2%7D%20%7D%7B%20%7B%20%7B20%7D%20%7D%5E%7B%7D%20%7D%20%20%3D%201)