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प्रश्न
Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse.
`x^2/100 + y^2/400 = 1`
उत्तर
The given equation is `x^2/100 + y^2/100 = 1` or `x^2/10^2 + y^2/20^2 = 1`
Here, the denominator of `y^2/400` is greater than the denominator of `x^2/100`.
Therefore, the major axis is along the y-axis, while the minor axis is along the x-axis.
On comparing the given equation with `x^2/b^2 + y^2/a^2 = 1` we obtain b = 10 and a = 20.
∴ `c = sqrt(a^2 - b^2) = sqrt(400 - 100) = sqrt(300) = 10sqrt3`
Therfore,
Coordinates of foci are (0, ± c) i.e. `(0, ±10sqrt3)`
Coordinates of vertices are (0, ±a) i.e. (0, ± 20).
Length of major axis = 2a = 2 x 20 = 40
Length of minor axis= 2b = 2 x 10 = 20
Eccentricity (e) = `c/a = (10sqrt3)/20 = sqrt3/2`
Length of latus rectum = `(2b^2)/a = (2 xx 100)/20 = 10`
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