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 earth moves around the sun is elliptical path. so by using the properties of ellipse \[{{r}_{1}}=(1+e)\,a\] and \[{{r}_{2}}=(1-e)\,a\] \[\Rightarrow \,a=\frac{{{r}_{1}}+{{r}_{2}}}{2}\] and \[{{r}_{1}}{{r}_{2}}=(1-{{e}^{2}})\,{{a}^{2}}\] where a = semi major axis b = semi minor axis e = eccentricity Now required distance = semi latusrectum \[=\frac{{{b}^{2}}}{a}\] \[=\frac{{{a}^{2}}(1-{{e}^{2}})}{a}=\frac{({{r}_{1}}{{r}_{2}})}{({{r}_{1}}+{{r}_{2}})/2}=\frac{2{{r}_{1}}{{r}_{2}}}{{{r}_{1}}+{{r}_{2}}}\]
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