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प्रश्न
Find the
- lengths of the principal axes.
- co-ordinates of the focii
- equations of directrics
- length of the latus rectum
- distance between focii
- distance between directrices of the ellipse:
3x2 + 4y2 = 12
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उत्तर
The equation of the ellipse is 3x2 + 4y2 = 12
i.e. `x^2/4 + y^2/3` = 1
Comparing with `x^2/"a"^2 + y^2/"b"^2` = 1, we get,
a2 = 4, b2 = 3
∴ a = 2, b = `sqrt(3)`
∴ a > b
i. Length of major axis = 2a = 2(2) = 4
Length of minor axis = 2b = `2sqrt(3)`
ii. Eccentricity = e = `sqrt("a"^2 - "b"^2)/"a"`
= `sqrt(4 - 3)/2`
= `1/2`
∴ ae = `2 xx 1/2` = 1
∴ coordinates of foci = (± ae, 0) = (± 1, 0).
iii. `"a"/"e" = 2/((1/2))` = 4
The equations of directrices are
x = `± "a"/"e"`
∴ x = ± 4
iv. Length of latus rectum = `(2"b"^2)/"a"`
= `(2 xx 3)/2`
= 3
v. Distance between foci = 2ae
= `2 xx 2 xx 1/2`
= 2
vi. Distance between directires = `(2"a")/"e"`
= `(2 xx 2)/((1/2))`
= 2 × 4
= 8
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