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(A) By the law of conservation of angular momentum,the angular momentum $L$ of the planet remains constant at all points in its orbit.At the perihelio

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$\frac{v_P}{v_A}=\frac{1+e}{1-e}$

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(A) By the law of conservation of angular momentum,the angular momentum $L$ of the planet remains constant at all points in its orbit.At the perihelion $P$ and aphelion $A$,the velocity vector is perpendicular to the position vector,so the angular momentum is given by $L = mvr$.Since $L_P = L_A$,we have $m v_P r_P = m v_A r_A$,which implies $\frac{v_P}{v_A} = \frac{r_A}{r_P}$.For an ellipse with semi-major axis $a$ and eccentricity $e$,the distance of the closest point (perihelion) is $r_P = a(1-e)$ and the distance of the farthest point (aphelion) is $r_A = a(1+e)$.Substituting these values,we get $\frac{v_P}{v_A} = \frac{a(1+e)}{a(1-e)} = \frac{1+e}{1-e}$.

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