Question
Download Solution PDFFind the equation of the hyperbola with vertices at (± 6, 0) and foci at (± 8, 0) ?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFCONCEPT:
The properties of a rectangular hyperbola
- 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:
- 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:
CALCULATION:
Here, we have to find the equation of the hyperbola with vertices at (± 6, 0) and foci at (± 8, 0)
By comparing the given vertices at (± 6, 0) with (± a, 0) we get
⇒ a = 6
Similarly, by comparing the given foci at (± 8, 0) with (± ae, 0) we get,
⇒ ae = 8
⇒ 6e = 8 ⇒ e = 4/3
As we know that,
⇒ a2e2 = a2 + b2
By substituting a2 = 36 and e2 = 16/9 in the above equation we get,
⇒ 36 × (16/9) = 36 + b2
⇒ b2 = 64 - 36 = 28
So, the required equation of hyperbola is
Hence, option A is the correct answer.
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