In each of the following find the equations of the hyperbola satisfying the given conditions
foci (0, ± 12), latus-rectum = 36.
Given: Foci (0, ±12), the latus-rectum = 36
To find: equation of the hyperbola
Formula used:
The standard form of the equation of the hyperbola is,
Coordinates of the foci for a standard hyperbola is given by (0, ±be)
Length of latus rectum is
According to the question:
be = 12
We know,
a2 = b2(e2 – 1)
⇒ 18b = 144 – b2
⇒ b2 + 18b – 144 = 0
⇒ b2 + 24b – 6b – 144 = 0
⇒ b(b + 24) – 6(b + 24) = 0
⇒ (b + 24)(b – 6) = 0
⇒ b = -24 or b = 6
Since b is a distance, and it can’t be negative
⇒ b = 6
⇒ b2 = 36
a2 = 18b
⇒ a2 = 18(6)
⇒ b2 = 108
Hence, equation of hyperbola is: