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Question: The largest and the shortest distance of the earth from the sun are \(r _ { 1 }\) and \(r _ { 2 }\),...

The largest and the shortest distance of the earth from the sun are r1r _ { 1 } and r2r _ { 2 }, its distance from the sun when it is at the perpendicular to the major axis of the orbit drawn from the sun

A

r1+r24\frac { r _ { 1 } + r _ { 2 } } { 4 }

B

r1r2r1+r2\frac { r _ { 1 } r _ { 2 } } { r _ { 1 } + r _ { 2 } }

C

2r1r2r1+r2\frac { 2 r _ { 1 } r _ { 2 } } { r _ { 1 } + r _ { 2 } }

D

r1+r23\frac { r _ { 1 } + r _ { 2 } } { 3 }

Answer

2r1r2r1+r2\frac { 2 r _ { 1 } r _ { 2 } } { r _ { 1 } + r _ { 2 } }

Explanation

Solution

The earth moves around the sun is elliptical path. so by using the properties of ellipse

r1=(1+e)ar _ { 1 } = ( 1 + e ) a and r2=(1−e)ar _ { 2 } = ( 1 - e ) a

⇒a=r1+r22\Rightarrow a = \frac { r _ { 1 } + r _ { 2 } } { 2 } and r1r2=(1−e2)a2r _ { 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 =b2a= \frac { b ^ { 2 } } { a }

=a2(1−e2)a=(r1r2)(r1+r2)/2=2r1r2r1+r2= \frac { a ^ { 2 } \left( 1 - e ^ { 2 } \right) } { a } = \frac { \left( r _ { 1 } r _ { 2 } \right) } { \left( r _ { 1 } + r _ { 2 } \right) / 2 } = \frac { 2 r _ { 1 } r _ { 2 } } { r _ { 1 } + r _ { 2 } }