Question

A planet revolves around sun in an elliptical orbit of eccentricigty e. Minor major axis

Answer 1

The shape of the orbit depends on the actual velocity of the body.
The orbit would have been perfectly circular if the planets had exactly the required minimum orbital velocity which is roughly given by-

V²= GM/R
where v= orbital velocity

G = universal gravitation constant,
G = 6.67×10^−11 N·(m/kg)²

M = mass of the planet that the body is orbiting, for Sun m ≈ 2 x 10^30 kg

R= Radius of orbit, for Earth, R ≈ 149.6 x 10^9 m

For earth, this velocity comes out to be about 29.8 km/s.

Remember that this is the minimum required velocity for Earth to continue orbiting the sun. Of course this is basic and rough formula without considering any other factors. At this speed, the minimum orbital velocity, the Earth would orbit the Sun in a circular orbit.

But our planet has more velocity than the minimum required. This makes the orbit elliptical ('e' (eccentricity) < 1). If the Earth had velocity equal to exactly the escape velocity (the velocity required to escape gravitational field of Sun), it would escape in a parabolic trajectory (e = 1) and if it had velocity even greater than this, it would move in a hyperbolic trajectory (e > 1).

Of course this is a highly debated topic and many theories have been suggested and many people may disagree with this explanation. Other factors include effects of other planets on earth, bulging of the earth and many other more complicated theories.

So, basically because the earth has the velocity greater than the minimum obital velocity required, corresponding to an elliptical orbit, it follows it. It will be very rare that any planet has exactly the velocity required for circular orbit. Moreover, if a body had exactly the velocity required for a circular orbit and it slowed down a bit, it would start falling into the sun. Therefore elliptical orbits are most common and generally stable.