Correct Answer - Option 1 :
\(\frac{{{x^2}}}{{{9}}} - \frac{{{y^2}}}{{{16}}} = 1\)
CONCEPT:
The properties of a rectangular hyperbola \(\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1\) are:
- Its centre is given by: (0, 0)
- Its foci are given by: (- ae, 0) and (ae, 0)
- Its vertices are given by: (- a, 0) and (a, 0)
- Its eccentricity is given by: \(e = \frac{{\sqrt {{a^2} + {b^2}} }}{a}\)
- Length of transverse axis = 2a and its equation is y = 0.
- Length of conjugate axis = 2b and its equation is x = 0.
- Length of its latus rectum is given by: \(\frac{2b^2}{a}\)
CALCULATION:
Here, we have to find the equation of the hyperbola whose foci are (± 5, 0) and the conjugate axis is of length 8.
By comparing the foci (± 5, 0) with (± ae, 0)
⇒ ae = 5
∵ Length of the conjugate axis is given by 2b
⇒ 2b = 8
⇒ b = 4
As we know that, eccentricity of a hyperbola is given by \(e = \frac{{\sqrt {{a^2} + {b^2}} }}{a}\)
⇒ a2e2 = a2 + b2
⇒ 25 = a2 + 16
⇒ a2 = 9
So, the equation of the required hyperbola is \(\frac{{{x^2}}}{{{9}}} - \frac{{{y^2}}}{{{16}}} = 1\)
Hence, option A is the correct answer.