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

Question

Dr. Abubacker · Accepted Answer

(c) 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 $r_{1}r_{2} = \left( 1 - e^{2} \right)a^{2}$
Where $a =$ semi major axis
$b =$ semi minor axis
$e =$ eccentricity
Now required distance = semi latusrectum $= \frac{b^{2}}{a}$
$= \frac{a^{2}(1 - e^{2})}{a} = \frac{(r_{1}r_{2})}{(r_{1} + r_{2})/2} = \frac{2r_{1}r_{2}}{r_{1} + r_{2}}$

Q. 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

Explanation

Reviewed by Dr. Abubacker

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