If the line \( x-2 y=12 \) is tangent to the ellipse \( \mathrm{P} ...

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If the line \( x-2 y=12 \) is tangent to the ellipse
\( \mathrm{P} \) \( \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \) at the point \( \left(3,-\frac{9}{2}\right) \), then the length of

W the latus rectum of the ellipse is:
(1) 5
(2) 9
(3) \( 8 \sqrt{3} \)
(4) \( 12 \sqrt{2} \)
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