An ellipse has semi-axes of length 6,33. The distance of a point P on this ellipse from one of the focus S is 92 , and the tangent at P to the ellipse cuts the major axis at T. Then, the length of ST is…………….

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An ellipse has semi-axes of length $6, 3 \sqrt{3}$ . The distance of a point $P$ on this ellipse from one of the focus $S$ is $\frac{9}{2}$ , and the tangent at $P$ to the ellipse cuts the major axis at $T$ . Then, the length of ST is…………….

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answer is 9.

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Detailed Solution

the tangent to an ellipse at any point bisects one pair of angles between the lines joining that point to the foci of the ellipse.
$\frac{S^{'} P}{SP} = \frac{S^{'} T}{ST} \Rightarrow \frac{(\frac{15}{2})}{\frac{9}{2}} = \frac{S^{'} T}{ST} \Rightarrow \frac{S^{'} S}{ST} = \frac{2}{3} \Rightarrow \frac{6}{ST} = \frac{2}{3}$

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