A) \[\frac{{{r}_{1}}+{{r}_{2}}}{4}\]
B) \[\frac{{{r}_{1}}{{r}_{2}}}{{{r}_{1}}+{{r}_{2}}}\]
C) \[\frac{2{{r}_{1}}{{r}_{2}}}{{{r}_{1}}+{{r}_{2}}}\]
D) \[\frac{{{r}_{1}}+{{r}_{2}}}{3}\]
Correct Answer: C
Solution :
The equation of a general conic is \[\frac{1}{r}=\frac{1}{l}(1+e\cos \theta )\]where e is eccentricity. For ellipse, turning points are at \[\theta =0{}^\circ \] and\[\theta =180{}^\circ \] giving \[{{r}_{\min }}={{r}_{2}}\] and \[{{r}_{\max }}={{r}_{1}}\] respectively \[\therefore \] \[\frac{1}{{{r}_{2}}}=\frac{1}{l}(1-e)\] and \[\frac{1}{{{r}_{1}}}=\frac{1}{l}(1+e)\] \[\therefore \] \[\frac{1}{{{r}_{2}}}+\frac{1}{{{r}_{1}}}=\frac{2}{l}\] or \[l=\frac{2{{r}_{1}}{{r}_{2}}}{{{r}_{1}}+{{r}_{2}}}\]
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