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Standard Equation of Ellipse

The equation ...

Question

The equation of the ellipse having foci (0, 1),(0, –1) and minor axis of length 1 is ___________.

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Solution

For an ellipse, length of minor axis is given = l and foci is given by = (0, $\pm$ 1)
i.e ellipse is of the form $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1; a < b$
i.e be = $\pm$ 1
Also, 2a = 1 i.e $\frac{1}{a}$ = 2
i.e a = $\frac{1}{2}$
and e = $\pm \frac{1}{b}$
$Since eccentricity e = \sqrt{1 - \frac{a^{2}}{b^{2}}} i . e e^{2} = 1 - \frac{a^{2}}{b^{2}} i . e \frac{1}{b^{2}} = 1 - \frac{1}{4 b^{2}} \frac{1}{b^{2}} + \frac{1}{4 b^{2}} = 1 i . e \frac{4 + 1}{4 b^{2}} = 1 i . e \frac{1}{b^{2}} = \frac{4}{5} ∴ \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 i . e \frac{x^{2}}{\frac{1}{4}} + \frac{4 y^{2}}{5} = 1 i . e 4 x^{2} + \frac{4 y^{2}}{5} = 1 i . e 20 x^{2} + 4 y^{2} = 5$

Hence, equation of ellipse is 20x² + 4y² = 5.

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