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Question
Identify the type of conic and find centre, foci, vertices, and directrices of the following:
`(x + 3)^2/225 + (y - 4)^2/64` = 1
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Solution
It is an hyperbola.
The transverse axis is parallell to x axis.
a2= 225, b2 = 64
a = 15, b = 8
c2 = a2 – b2
= 225 + 64
c2 = 289
c = 17
ae = 17
5e = 17
e = `17/15`
Centre (h, k) = (– 3, 4)
Vertices (h ± a, k) = (– 3 ± 15, 4)
= (– 3 + 15, 4) and (– 3 – 15, 4)
= (12, 4) and (– 18, 4)
Foci (h ± c, k) = (– 3 ± 17, 4)
= (– 3 + 17, 4) and (– 3 – 17, 4)
= (14, 4) and (– 20, 4)
Directrix x = `+- "a"/"e" + "h"`
= `+- 15/(17/5) - 3`
= `+- 225/17 - 3`
x = `225/17 - 3` and x = `- 225/17 - 3`
= `174/17` and = `(- 276)/17`
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