parabola part 4 doubts

Try finding the centre and radius of the circle. (Centre is negative and passes through origin). Also, find the point of intersection of circle and parabola, which will give 2 values. One is zero and other in terms of lambda. But from the figure, it can be deduced that both intersect at only origin. Hence, equate the one value in terms of lambda to zero and get the value of Lambda.

Why tangent? It would take you more time to solve the eqn.

Completion of squares of eqn of circle will give you centre and radius. (-2L,0) and radius(2L)

Solve for points of intersection of circle and parabola, to get the solution:
x=0,-8-4L

As only point lies where both intersect i.e. at origin. So -8-4L=0

And the value of L from here is not far off...