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The minimum area of the triangle formed by any tangent to the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ with the coordinate axes is
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$a b$
Equation of tangent at $(a \cos \theta, b \sin \theta)$ to the ellipse is
$\frac{x}{a} \cos \theta+\frac{y}{b} \sin \theta=1$
Coordinates of $P$ and $Q$ are $\left(\frac{\mathrm{a}}{\cos \theta}, 0\right)$ and $\left(0, \frac{\mathrm{b}}{\sin \theta}\right)$, respectively.
Area of $\Delta \mathrm{OPQ}=\frac{1}{2}\left|\frac{\mathrm{a}}{\cos \theta} \times \frac{\mathrm{b}}{\sin \theta}\right|=\frac{\mathrm{ab}}{|\sin 2 \theta|}$
$\frac{x}{a} \cos \theta+\frac{y}{b} \sin \theta=1$
Coordinates of $P$ and $Q$ are $\left(\frac{\mathrm{a}}{\cos \theta}, 0\right)$ and $\left(0, \frac{\mathrm{b}}{\sin \theta}\right)$, respectively.
Area of $\Delta \mathrm{OPQ}=\frac{1}{2}\left|\frac{\mathrm{a}}{\cos \theta} \times \frac{\mathrm{b}}{\sin \theta}\right|=\frac{\mathrm{ab}}{|\sin 2 \theta|}$
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