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

• zorro
In summary, when solving for the intersection of the circle x2 + y2 =1 and the parabola y=ax2 - b, provided a>b>1, the solution involves finding the values of b and a that satisfy the equations. The key step in the solution is understanding that b/a will always be less than one, which is necessary for the parabola to intersect with the circle. This is because a must be larger than b in order for the parabola to intersect the y-axis below the origin.
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 ?

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.

1. What is the equation for a circle?

The equation for a circle is x2 + y2 = r2, where (x,y) represents any point on the circle and r is the radius.

2. What is the equation for a parabola?

The equation for a parabola is y = ax2 + bx + c, where a, b, and c are constants and a ≠ 0.

3. How do you prove circle-parabola intersection for a>b>1?

To prove circle-parabola intersection for a>b>1, we can substitute y = ax2 - b into the equation for a circle, x2 + y2 = 1, and solve for x. This will give us two values for x, which we can then substitute back into the equation for the parabola to find the corresponding y values. If these points lie on both the circle and the parabola, then there is an intersection between the two.

4. Can there be more than two points of intersection between a circle and a parabola?

Yes, there can be more than two points of intersection between a circle and a parabola. In fact, there can be up to four points of intersection, depending on the specific values of a, b, and c in the equation for the parabola.

5. What does the condition a>b>1 signify in the equation for circle-parabola intersection?

The condition a>b>1 signifies that the parabola is narrower, or more vertically stretched, than the circle. This condition is necessary for there to be an intersection between the two curves, as a wider parabola would not intersect with the circle at any point.

Replies
5
Views
3K
Replies
2
Views
1K
Replies
3
Views
2K
Replies
5
Views
1K
Replies
1
Views
1K
Replies
3
Views
2K
Replies
27
Views
8K
Replies
3
Views
1K
Replies
11
Views
3K
Replies
4
Views
1K