A planet is revolving round the sun. Its distance from the sun at Apogee is $r_{A}$ and that at Perigee is $r_{P}$ . The mass of planet and sun is $m$ and $M$ respectively, $v_{A}$ and $v_{P}$ is the velocity of planet at Apogee and Perigee respectively and $T$ is the time period of revolution of planet round the sun.
Then consider the following:
(a) $T^{2} = \frac{π^{2}}{2 G m} {(r_{A} + r_{P})}^{2}$
(b) $T^{2} = \frac{π^{2}}{2 G M} {(r_{A} + r_{P})}^{3}$
(c) $v_{A} r_{A} = v_{P} r_{P}$
(d) $v_{A} < v_{P}, r_{A} > r_{P}$
Choose the correct statements from (a), (b), (c) and (d) :

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a , b , c

a , b , d

all

b , c , d

answer is C.

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Detailed Solution

$T^{2} = \frac{4 π^{2}}{G M} a^{3} T^{2} = \frac{4 π^{2}}{G M} {(\frac{r_{A} + r_{P}}{2})}^{3} = \frac{π^{2}}{2 G M} {(r_{A} + r_{P})}^{3}$

$m v r = constant \Rightarrow v r = constant$

$∴ v_{A} r_{A} = v_{P} r_{P} ∴ r_{A} > r_{P}; ∴ v_{A} < v_{P}$

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