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Ends of major axis lie on y-axis. ∴ Equation of ellipse in the standard form is Ends of major axis are Ends of minor axis are Hence, from (i), the equation of ellipse is Also, ∴ The foci are - Wired Faculty

Question

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Find the equation of an ellipse that satisfies the following conditions
Ends of major axis at $space left parenthesis 0 comma space plus-or-minus square root of 13 right parenthesis comma$ ends of minor axis at $space space left parenthesis plus-or-minus 2 comma space 0 right parenthesis$ . Also find the co-ordinate of foci.

Solution

Ends of major axis $left parenthesis 0 comma space plus-or-minus square root of 13 right parenthesis$ lie on y-axis.
∴ Equation of ellipse in the standard form is $space space straight y squared over straight a squared plus straight x squared over straight b squared equals 1$
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Ends of minor axis are $left parenthesis plus-or-minus straight b comma space 0 right parenthesis space left right arrow space left parenthesis plus-or-minus 2 comma space 0 right parenthesis space rightwards double arrow space straight b space equals space 2$
Hence, from (i), the equation of ellipse is $space space fraction numerator straight y squared over denominator left parenthesis square root of 13 right parenthesis squared end fraction plus straight x squared over 2 squared equals 1 space or space straight y squared over 13 plus straight x squared over 4 equals 1$
Also, $straight c squared equals straight a squared minus straight b squared equals 13 minus 14 space rightwards double arrow space straight c squared equals 9 space space or space straight c equals 3$
∴ The foci are $left parenthesis 0 comma space plus-or-minus straight c right parenthesis space or space left parenthesis 0 comma space plus-or-minus 3 right parenthesis.$

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