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The largest and the shortest distance of...

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 perpendicular to the major axis of the orbit drawn from the sun

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Since `e = c/a rArr c =ea " " therefore r_1 = a+c=a+ea=a(1+e)`
` r_2 = a -c =a -ea = a (1-e)`
Required distance `L = (b^2)/(a)` (semi latus rectum )
we have `r_1+r_2 = 2a` and `r_-r_2 =a(1+e) -a(1-e) = 2ae rArr e = (r_1-r_2)/(r_1+r_2)`
Also `b^2 = a^2 (1-e^2)`
` therefore L = (b^2)/(a) = (a^2 (1-e^2))/(a) = a (1-e^2)`
` = ((r_1+r_2)/(2)) [1-((r_1-r_2)/(r_1+r_2))^2]`
on simplification , `L = (2r_1r_2)/(r_1+r_2) "or " 2/L = (1)/(r_1) + (1)/(r_2)`
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