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The largest a...

Question

The largest and the shortest distance of the earth from the sun are $r_{1}$ and $r_{2}$ , its distance from the sun when it is at the perpendicular to the major axis of the orbit drawn from the sun

A

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C

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Solution

The correct option is C

The earth moves around the sun is elliptical path. So by using the properties of ellipse
$r_{1} = (1 + e) a a n d r_{2} = (1 - e) a$
$\Rightarrow a = \frac{r_{1} + r_{2}}{2} a n d r_{1} r_{2} = (1 - e^{2}) a^{2}$

where a=semi major axis
b=semi minor axis
e=eccentricity

Now required distance=semi latusrectum= $\frac{b^{2}}{a}$
= $\frac{a^{2} (1 - e^{2})}{a} = \frac{(r_{1} r_{2})}{\frac{(r_{1} + r_{2})}{2}} = \frac{2 r_{1} r_{2}}{r_{1} + r_{2}}$

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