Proving Circle-Parabola Intersection: a>b>1 | x2 + y2 = 1, y = ax2 - b

[zorro]

Homework Statement

For a > 0, prove that the circle x2 + y2 =1 and the parabola y=ax2 - b
intersect at four distinct points, provided a>b>1.

2. The attempt at a solution

This is the solution given in my book.
Since a>0, by figure -b<-1 i.e. b>1
also when y=0
x2=b/a (from the equation of the parabola)

The step which I did not understand is this-

Therefore, b/a <1 i.e.
b<a
Hence a>b>1

Please explain how is b/a <1 ?

 

[FLUndergrad]

b/a will always be less than one because a is bigger than b. a has to be bigger than b because otherwise the parabola will become too wide to intersect with the circle relative to how low on the y-axis its vertex is. It will always intersect the y-axis under the origin because of the parameters of the problem. Dunno if that's the kind of explanation you were looking for but I hope it helps.