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Question
Identify the type of conic and find centre, foci, vertices, and directrices of the following:
`(x + 1)^2/100 + (y - 2)^2/64` = 1
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Solution
It is an ellipse.
The major axis is parallel to the x-axis.
a2 = 100, b2 = 64
a = 10, b = 8
c2 = a2 – b2
= 100 – 64 = 36
c = 6
ae = 6
10e = 6
e = `6/10 = 3/5`
Centre (h, k) = (– 1, 2)
Vertices (h ± a, k) = (– 1 ± 10, 2)
= (– 1 + 10, 2) and (– 1 – 10, 2)
= (9, 2) and (– 11, 2)
Foci (h ± c, k) = (– 1 ± 6, 2)
= (– 1 + 6, 2) and (– 1 – 6, 2)
= (5, 2) and (– 7, 2)
Directrix x = `+- "a"/"e" + "h"`
= `+- 10/(3/5) - 1`
= `+- 50/3 - 1`
x = `50/3 - 1` and x = `50/(-3) - 1`
= `47/3` and `(- 53)/5`
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