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Question

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 36x2+4y2=144

Solution
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The given equation is 36x2+4y2=144
It can be written as
36x2+4y2=144
x24+y236=1
x222+y262=1 ...(1)
Here the donominator of y262 is greater than the denominator of x222
Therefore, the major axis is along the y-axis while the minor axis is along the x-axis.
On comparing equation (1) with x2b2+y2a2=1, we obtain b=2 and a=6
ae=c=a2b2=364=32=42
Therefore, the coordinates of the foci are (0,±42)
The coordinates of the vertices are (0, ±6)
Length of major axis =2a=12
Length of minor axis =2b=4
Eccentricity e= ca=426=223
Length of latus rectum = 2b2a=2×46=43

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