4 5 6 7 ............?....?.............................................................

There are 4 question so i will solve them one by one first we proceed with 4-)There is a well know theorem in hyperbola that any tangent to hyperbola makes constant area with two asymptotes and that area is equal to AB.Now area is given a^2tan which is equal to ab .Now b=a(e^2-1)^1/2 so ab can be written a^2(e^2-1)^1/2.Now compare a^2tan=a^2(e^2-1)^1/2 or tan=(e^2-1)^1/2 or tan^2=e^2-1 or tan^2+1=e^2 finally e=sec.Hence answer is ecentricity=sec(lamda)