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Find the (i) lengths of major and minor axes, (ii) coordinates of the vertices, (iii) coordinates of the foci, (iv) eccentricity, and (v) length of the latus rectum of the following ellipse.

16x2 + 25y2 = 400

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Divide both sides by 400

16/400 x2 + 25/400 y2 = 1

or x2/25 + y2/16 = 1

For general form of ellipse:

x2/a2 + y2/b2 = 1 ….(1)

On comparing given equation with (1), we get

a = 5 and b = 4

Then,

c2 = a2 – b2

c2 = 25 – 16 = 9

or c = 3

Now,

(i) Lengths of major and minor axes:

Major axis: 2a = 10 units

Minor axis : 2b = 8 units

(ii) Coordinates of the vertices: (±a, 0) = (±5, 0)

(iii) Coordinates of the foci: (±c, 0) = (±3, 0)

(iv) Eccentricity: e = c/a = 3/5

(v) Length of the latus rectum: 2b2/a = 2(4)2/5 = 32/5 units

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