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The smallest possible positive slope of a line whose y-intercept is 5 and which has a common point with the ellipse $9 x^{2}+16 y^{2}=144$ is-
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ellipse $\frac{x^{2}}{16}+\frac{y^{2}}{9}=1$
Any tangent $\frac{x \cos \theta}{4}+\frac{y \sin \theta}{3}=1$
$\begin{array}{c}
\text { y intercept }=5 \Rightarrow \sin \theta=\frac{3}{5} \quad ; \theta \in\left(\frac{\pi}{2}, \pi\right) \\
\Rightarrow \cos \theta=-\frac{4}{5} \\
\text { tangent } \Rightarrow-\frac{x}{5}+\frac{y}{5}=1 \Rightarrow \text { slope }=1
\end{array}$
Any tangent $\frac{x \cos \theta}{4}+\frac{y \sin \theta}{3}=1$
$\begin{array}{c}
\text { y intercept }=5 \Rightarrow \sin \theta=\frac{3}{5} \quad ; \theta \in\left(\frac{\pi}{2}, \pi\right) \\
\Rightarrow \cos \theta=-\frac{4}{5} \\
\text { tangent } \Rightarrow-\frac{x}{5}+\frac{y}{5}=1 \Rightarrow \text { slope }=1
\end{array}$
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