A bar of length 20 units moves with its ends on two fixed straight lines at right angles. A point $P

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A bar of length 20 units moves with its ends on two fixed straight lines at right angles. A point $P$ marked on the bar at a distance of 8 units from one end describes a conic whose eccentricity is:

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Sol.(d)$$\begin{matrix} & \ (AB)^{2} = \left( \frac{5h}{3} ight)^{2} + \left( \frac{5k}{2} ight)^{2} = (20)^{2} \ \Rightarrow & \frac{x^{2}}{9} + \frac{y^{2}}{4} = 16 \ \Rightarrow & \frac{x^{2}}{144} + \frac{y^{2}}{64} = 1 \ & 64 = 144\left( 1 - e^{2} ight) \ \Rightarrow & e^{2} = 1 - \frac{4}{9} = \frac{5}{9} \Rightarrow e = \frac{\sqrt{5}}{3} \end{matrix}$$

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