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The value of a for the ellipse (x^2)/(a^...

The value of `a` for the ellipse `(x^2)/(a^2)+(y^2)/(b^2)=1,(a > b),` if the extremities of the latus rectum of the ellipse having positive ordinates lie on the parabola `x^2=2(y-2)` is ___

Text Solution

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`(+-ae,b^(2)//a)` are extermities of the latus rectum having positive ordinates. Then.
`a^(2)e^(2)=-2((b^(2))/(a)-2) " "(1)`
But `b^(2)=a^(2)(1-e^(2))" "(2)`
Therefore, from (1) and (2), we get
`a^(2)e^(2)-2ae^(2)-4=0`
or `ae^(2)(a-2)+2(a-2)=0`
`:. (ae^(2)+2)(a-2)=0`
Hence, a=2
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