This is was an exercise on my exam of classical mechanics which I couldn't solve. The question goes as follows:
Imagine a comet moving on a parabolic orbit in the plane of the earth. Take the earth's orbit as a circle with radius R and the sun in the centre. The shortest distance that the comet comes to the sun is $\beta R$ with $\beta<1$. So the comet is on the inside of the earth's orbit for a certain time. Prove that the time the comet spends in the circle is $$ \Delta t = \frac{1}{3\pi}\sqrt{2(1-\beta)}(1+2\beta) \times \text{year} $$
Hints: $$ \int \cos^{-4}x\,dx= \tan{x}+ \frac{\tan^3{x}}{3} + C, $$ $$ \cos{2x}=1-2\sin^2{x} $$
I tried calculating the path of the comet with the polar equations of a hyperbola, but I have no clue how to solve this.