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  • The largest and the shortest distances of the earth from the sun are <img alt="\mathrm{r_1 \: and \: r_2}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7Br_1%20%5C%3A%20and%20%5C%

The largest and the shortest distances of the earth from the sun are \mathrm{r_1 \: and \: r_2}. It's distance from the sun when it is at the perpendicular to the major axis of the orbit drawn from the sun is
 

Option: 1

\mathrm{\frac{r_1+r_2}{4}}


 


Option: 2

\mathrm{\frac{\mathrm{r}_1 \mathrm{r}_2}{\mathrm{r}_1+\mathrm{r}_2}}
 


Option: 3

\mathrm{\frac{2 \mathrm{r}_1 \mathrm{r}_2}{\mathrm{r}_1+\mathrm{r}_2}}
 


Option: 4

\mathrm{\frac{r_1+r_2}{3}}


Answers (1)

best_answer

The equation of a general conic is \mathrm{\frac{1}{\mathrm{r}}=\frac{1}{\ell}(1+\mathrm{e} \cos \theta)}, where \mathrm{e} is eccentricity. For ellipse, turning points are at \mathrm{\theta=0, \: and \: \theta=180\: giving \: r_{\min }=r_2}, and \mathrm{r_{\max }=r_1} respectively.

\mathrm{ \therefore \frac{1}{\mathrm{r}_2}=\frac{1}{\ell}(1+\mathrm{e}) \text { and } \frac{1}{\mathrm{r}_1}=\frac{1}{\ell}(1-\mathrm{e}) }

\mathrm{\text { Add } \frac{1}{\mathrm{r}_2}+\frac{1}{\mathrm{r}_1}=\frac{2}{\ell} ; 1=\frac{2 \mathrm{r}_1 \mathrm{r}_2}{\mathrm{r}_1+\mathrm{r}_2} }

Hence option 3 is correct.




 

Posted by

Divya Prakash Singh

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