Statement-l :
The major and minor axes of the ellipse respectively are $5 x^{2} + 9 y^{2} - 54 y + 36 = 0$ are $6$ and $10$ .

Statement-II :
The equation can be expressed as $5 x^{2} + 9 (y - 3)^{2} = 45$

Both Statement I and Statement II are correct and Statement II is the correct explanation of Statement I.

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Both Statement I and Statement II are correct and Statement II is NOT the correct explanation of Statement I.

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Statement I is correct and Statement II is incorrect.

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Statement I is incorrect and Statement II is correct.

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Solution

The correct option is D Statement I is incorrect and Statement II is correct.
The given equation is:
$5 x^{2} + 9 y^{2} - 54 y + 36 = 0 \Rightarrow 5 x^{2} + 9 y^{2} - 54 y + 81 - 45 = 0 \Rightarrow 5 x^{2} + {(3 y - 9)}^{2} = 45 \Rightarrow \frac{5 x^{2}}{45} + \frac{{9 (y - 3)}^{2}}{45} = 1 \Rightarrow \frac{x^{2}}{9} + \frac{{(y - 3)}^{2}}{5} = 1$
$∴ 3$ and $\sqrt{5}$ are major and minor axis, respectively.

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