There are public and private transport service between Ajmer and Delhi. Suppose that the private transport fare between the two cities come down. How will this affect the demand curve for public transport between the two cities?

Dear student 
If the prices of private transport between the two cities decreases the demand for private transport will increase and the demand for public transport will decrease as now private transport is cheaper than public transport.
The demand curve for public transport will shift towards the left at the same price i.e. the demand for public transport will decrease even though the price of public transport is same as before. This shift is because of the decrease in the price of private transport.

regards