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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)^2/289` = 1
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Solution
It is an ellipse.
The major axis is parallel to y axis
a2 = 289, b2 = 225
a = 17, b = 15
c2 = a2 – b2
= 289 – 225 = 64
c = 8
ae = 8
17e = 8
e = `8/17`
Vertices (h, ±a + k)
= (3, 17 + 4) and (3, – 17 + 4)!
= (3, 21) and (3, – 13)
Foci (h + 0, ± c + k)
= (3, 8 + 4) and (3, – 8 + 4)
= (3, 12) and (3, – 4)
Directrices y = `+- "a"/"e" + "k"`
= `+- 17/(8/17) + 4`
= `+- 289/8 + 4`
= `289/8 + 4` and `- 289/8 + 4`
= `(289 + 32)/8` and `(- 289 + 32)/8`
= `321/8` and `- 257/8`
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