An asteroid is orbiting the Sun in a circular orbit of radius 4AU. Calculate the ratio of its angular diameters at opposition and quadrature.
I have tried using the idea of elongation but it does not work for me.
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Here is a little picture. In such questions of geometric orbits, a picture helps greatly.
The Earth is at point B, and the asteroid at opposition is at C, but at quadrature it is at point D. Your first task is to work out the distances BC and BD.
Then the ratio of angular diameters is just the reciprocal of the ratio of distances.
This makes a couple of assumptions: the angular diameter is small (which it is) and the orbits are circlar (which is given for the asteroid and is a good approximation for the Earth.