question_answer

The largest and shortest distances of the earth from the sun are \[{{r}_{1}}\] and \[{{r}_{2}}\] respectively. What is the distance from the sun when it is perpendicular to the major axis of the orbit from the sun?

Answer:

                Let R be the distance of the earth from the sun in the position perpendicular to the major axis of its elliptical orbit.                 Using the property of an ellipse, we can write                 \[\frac{1}{R}+\frac{1}{R}=\frac{1}{{{r}_{1}}}+\frac{1}{{{r}_{2}}}\]                 or                            \[\frac{2}{R}=\frac{{{r}_{1}}+{{r}_{2}}}{{{r}_{1}}{{r}_{2}}}\]                 \[\therefore \]  \[R=\frac{2{{r}_{1}}{{r}_{2}}}{{{r}_{1}}+{{r}_{2}}}\].