Let A be the area of triangle formed by any tangent to the curve xy = 4 cosec²θ, θ ≠ nπ, nI and the co-ordinate axis. The minimum value of A is :- (1) 4 (2) 8 (3) 2 (4) 16

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$$\begin{gathered} \mathrm{xy}=4{\mathrm{cosec}}^2\mathrm{\theta } ;\left(\mathrm{i}\right) \\ \mathrm{A} \mathrm{general} \mathrm{point} \mathrm{on} \mathrm{line} \mathrm{will} \mathrm{be} \left(\mathrm{acosec\theta },\mathrm{bcosec\theta }\right) \\ \mathrm{where} \mathrm{ab}=4 \\ \mathrm{Differenting} \mathrm{both} \mathrm{sides} \mathrm{with} \mathrm{respect} \mathrm{to} \mathrm{x}, \mathrm{we} \mathrm{get}: \\ \mathrm{x}.\frac{\mathrm{dy}}{\mathrm{dx}}+\mathrm{y}.\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{x}\right)=0 \\ \Rightarrow \mathrm{x}.\frac{\mathrm{dy}}{\mathrm{dx}}+\mathrm{y}=0 \\ \Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=-\frac{\mathrm{y}}{\mathrm{x}}=-\frac{\mathrm{bcosec\theta }}{\mathrm{acosec\theta }}=-\frac{\mathrm{b}}{\mathrm{a}}=\mathrm{Slope}\left(\mathrm{m}\right) \mathrm{of} \mathrm{tangent} \\ \mathrm{Equation} \mathrm{of} \mathrm{tangent} \mathrm{is}: \\ \mathrm{y}-{\mathrm{y}}_1=\mathrm{m}\left(\mathrm{x}-{\mathrm{x}}_1\right) \\ \Rightarrow \mathrm{y}-\mathrm{bcosec\theta }=-\frac{\mathrm{b}}{\mathrm{a}}\left(\mathrm{x}-\mathrm{acosec\theta }\right) \\ \Rightarrow \mathrm{ay}-\mathrm{abcosec\theta }=-\mathrm{bx}+\mathrm{abcosec\theta } \\ \Rightarrow \mathrm{ay}-4\mathrm{cosec\theta }=-\mathrm{bx}+4\mathrm{cosec\theta } \left[\mathrm{As} \mathrm{ab}=4\right] \\ \Rightarrow \mathrm{bx}+\mathrm{ay}=8\mathrm{cosec\theta } \\ \Rightarrow \frac{\mathrm{bx}}{8\mathrm{cosec\theta }}+\frac{\mathrm{ay}}{8\mathrm{cosec\theta }}=1 \\ \Rightarrow \frac{\mathrm{x}}{\left({\displaystyle \frac{8\mathrm{cosec\theta }}{\mathrm{b}}}\right)}+\frac{\mathrm{y}}{\left(\frac{8\mathrm{cosec\theta }}{\mathrm{a}}\right)}=1 \\ \Rightarrow \mathrm{X}-\mathrm{axis} \mathrm{Intercept}=\mathrm{A}=\frac{8\mathrm{cosec\theta }}{\mathrm{b}} \\ \Rightarrow \mathrm{Y}-\mathrm{axis} \mathrm{Intercept}=\mathrm{B}=\frac{8\mathrm{cosec\theta }}{\mathrm{a}} \\ \mathrm{Area}=\frac{1}{2}\left|\mathrm{AB}\right| \\ =\frac{1}{2}\left|\left(\frac{8\mathrm{cosec\theta }}{\mathrm{b}}\right)\left(\frac{8\mathrm{cosec\theta }}{\mathrm{a}}\right)\right| \\ =\frac{64}{2\mathrm{ab}}\left|{\mathrm{cosec}}^2\mathrm{\theta }\right| \\ \mathrm{Minimum} \mathrm{value} \mathrm{of} {\mathrm{cosec}}^2\mathrm{\theta }=1 \\ \Rightarrow {\mathrm{Area}}_{\min }=\frac{64}{2\mathrm{ab}}=\frac{64}{2\times 4}=8\to \mathrm{Option}\left(2\right) \end{gathered}$$

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