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Question
Identify the type of conic and find centre, foci, vertices, and directrices of the following:
`(y - 2)^3/25 + (x + 1)^2/16` = 1
Solution
It is an hyperbola.
The transverse axis is parallell to y axis.
a2 = 25, b2 = 16
a = ± 5, b = 4
c2 = a2 + b2
= 25 + 16
= 41
c = `sqrt(41)`
ae = `sqrt(41)`
5e = `sqrt(41)`
e = `sqrt(41)/5`
Centre (h, k) = (– 1, 2)
Vertices (h, ± a + k) = (– 1, ± 5 + 2)
= (– 1, 5 + 2) and (– 1, – 5 + 2)
= (– 1, 7) and (– 1, – 3)
Foci (h, ± c + k) = `(- 1 +- sqrt(41) + 2)`
= `(-1, sqrt(41) + 2)` and `(-1, - sqrt(41) + 2)`
Directrix x = `+- "a"/"e" + "k"`
= `+- 5/(sqrt(41)/5) + 2`
= `+- 25/sqrt(41) + 2`
y = `25/sqrt(41) + 2` and y = `- 25/sqrt(41) + 2`
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