The mean radius of the earth's orbit around the sun is $1.5 \times 10^{11} \ m$. The mean radius of the orbit of mercury around the sun is $6 \times 10^{10} \ m$. The mercury will rotate around the sun in

The figure shows the motion of a planet around the sun in an elliptical orbit with the sun at the focus. The shaded areas $A$ and $B$ are also shown in the figure,which can be assumed to be equal. If ${t_1}$ and ${t_2}$ represent the time taken for the planet to move from $a$ to $b$ and $d$ to $c$ respectively,then:

The figure shows the motion of a planet around the sun in an elliptical orbit with the sun at the focus. The shaded areas are equal. If $t_1$ and $t_2$ represent the time taken for the planet to move from $A$ to $B$ and $C$ to $D$ respectively,then:

Kepler discovered

$A$ planet revolving in an elliptical orbit has:
$(A)$ a constant velocity of revolution.
$(B)$ the least velocity when it is nearest to the sun.
$(C)$ its areal velocity is directly proportional to its velocity.
$(D)$ areal velocity is inversely proportional to its velocity.
$(E)$ a trajectory such that the areal velocity is constant.
Choose the correct answer from the options given below:

Two planets $A$ and $B$ of equal mass have periods of revolution $T_{A}$ and $T_{B}$ such that $T_{A} = 2 T_{B}$. These planets are revolving in circular orbits of radii $r_{A}$ and $r_{B}$ respectively. Which of the following is the correct relationship between their orbital radii?