The first of the ellipse $25 x^{2} + 4 y^{2} + 100 x - 4 y + 100 = 0$ are

(\frac{5 \pm \sqrt{21}}{10}, - 2)

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(- 2, \frac{5 \pm \sqrt{21}}{10})

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(\frac{2 \pm \sqrt{21}}{10}, - 2)

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(- 2, \frac{2 \pm \sqrt{21}}{10})

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Solution

The correct option is B $(- 2, \frac{5 \pm \sqrt{21}}{10})$
Given, $25 x^{2} + 4 y^{2} + 100 x - 4 y = - 100$
$\Rightarrow 25 (x^{2} + 4 x) + 4 (y^{2} - y) = - 100$
$\Rightarrow 25 {(x + 2)}^{2} + 4 {(y - \frac{1}{2})}^{2} = 1$
$\frac{{(x + 2)}^{2}}{{(\frac{1}{5})}^{2}} + \frac{{(y - \frac{1}{2})}^{2}}{{(\frac{1}{2})}^{2}} = 1, b > a$
$e = \sqrt{\frac{\frac{1}{4} - \frac{1}{25}}{\frac{1}{4}}} = \frac{\sqrt{21}}{5}$
$s, s^{'} = (α, β \pm b e) = (- 2, \frac{1}{2} \pm \frac{\sqrt{21}}{10})$

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