Question
Download Solution PDFFind the equation of the hyperbola with vertices at (0, ± 5) and foci at (0, ± 8) ?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFCONCEPT:
The properties of a vertical hyperbola
- Its centre is given by: (0, 0)
- Its foci are given by: (0, - ae) and (0, ae)
- Its vertices are given by: (0, - a) and (0, a)
- Its eccentricity is given by:
- Length of transverse axis = 2a and its equation is x = 0.
- Length of conjugate axis = 2b and its equation is y = 0.
- Length of its latus rectum is given by:
CALCULATION:
Here, we have to find the equation of the hyperbola with vertices at (0, ± 5) and foci at (0, ± 8)
By comparing the given vertices at (0, ± 5) with (0, ± a) we get
⇒ a = 5
Similarly, by comparing the given foci at (0, ± 8) with (0, ± ae) we get,
⇒ ae = 8
⇒ 5e = 8 ⇒ e = 8/5
As we know that,
⇒ a2e2 = a2 + b2
By substituting a2 = 25 and e2 = 64/25 in the above equation we get,
⇒ 25 × (64/25) = 25 + b2
⇒ b2 = 64 - 25 = 39
So, the required equation of hyperbola is
Hence, option B is the correct answer.
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