The largest and the shortest distance of the earth from are r1 and r2. It’s distance from the sun when it is perpendicular to the major-axis of the orbit drawn from the sun.

  • Option 1)

    \left ( \frac{r1+r2 }{4}\right )

  • Option 2)

    \left ( \frac{r1+r2 }{r1-r2}\right )

  • Option 3)

    \left ( \frac{2r1r2}{r1+r2} \right )

  • Option 4)

    \left ( \frac{r1+r2}{3} \right )

 

Answers (1)
A Aadil

As we learnt in 

Velocity of planet in terms of Eccentricity -

V_{a}=\sqrt{\frac{GM}{a}\left ( \frac{1-e}{1+e} \right )}

V_{p}=\sqrt{\frac{GM}{a}\left ( \frac{1+e}{1-e} \right )}

V_{A}= Velocity of planet at apogee

V_{p}= Velocity of perigee

- wherein

Eccentricity (e) = \frac{c}{a}

r_{p}=a-c

r_{a}=a+c

 

The position of a particle moving in an elliptical orbit is represented as

r=\frac{l}{1+e \cos \Theta }

l is perpendicular distance of particle from focus and e is eccentricity of ellipse

r_{1}=\frac{l}{1-e}\ \: \: and\ \: \: \: r_{2}=\frac{l}{1+e}

\Rightarrow 1-e=\frac{l}{r_{1}}\ \: \: and\ \: \: 1+e = \frac{l}{r_{2}}

\Rightarrow 2=l\left ( \frac{1}{r_{1}} + \frac{1}{r_{2}}\right )\Rightarrow l=\frac{2r_{1}r_{2}}{r_{1}+r_{2}}


Option 1)

\left ( \frac{r1+r2 }{4}\right )

This is incorrect option

Option 2)

\left ( \frac{r1+r2 }{r1-r2}\right )

This is incorrect option

Option 3)

\left ( \frac{2r1r2}{r1+r2} \right )

This is correct option

Option 4)

\left ( \frac{r1+r2}{3} \right )

This is incorrect option

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