The equation of the latusrecta of the ellipse $9 x^{2} + 4^{2} - 18 x - 8 y - 23 = 0$ are

y = \pm \sqrt{5}

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x = \pm \sqrt{5}

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y = 1 \pm \sqrt{5}

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x = 1 \pm \sqrt{5}

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Solution

The correct option is D $y = 1 \pm \sqrt{5}$
equation of ellipse is ${9 x}^{2} + {4 y}^{2} - 18 x - 8 y - 23 = 0$
$\Rightarrow (9 x^{2} - 18 x + 9) + (4 y^{2} - 8 y + 4) - 23 - 9 - 4 = 0$
$\Rightarrow 9 (x - 1)^{2} + 4 (y - 1)^{2} = 36$
$\Rightarrow \frac{(x - 1)^{2}}{4} + \frac{(y - 1)^{2}}{9} = 1$
So, equation of latus recta is $(y - 1) = \pm b e$
$y = 1 \pm \sqrt{b^{2} - a^{2}} \Rightarrow y = 1 \pm \sqrt{5}$

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