A Little Challenge To Stave Off Boredom?

Question

A circle, with its circumference touching the
graph of y = x^3 at the origin, also touches
only one other point of the graph above the
X-axis. What is the exact radius of the circle?

To She-Nerd : note that the circumference of the circle, not the centre, is on the origin.

To zanti3 : bad word choice on my part - yes, touching means tangent.

Answer 1

By numerical approximation, the radius is 0.877384. Now, let's do it the exact way. The equation of the circle of radius k tangent to the origin is:

k - √(k² - x²)

so all we have to do is to find a double root in the equation:

x³ - k + √(k² - x²) = 0

so, hold on

Addendum: Terrific, the exact value of k is (2/(3^(3/4)), which comes to about 0.877383, so my numerical approximation wasn't too shabby. The way to find this is to first expand the equation above and then divide it by x², which represents the double root at the origin. Then we investigate what values a, b, c must be in order for following for this to be true:

x^4 - 2kx + 1 = (x - a)²(x² + bx + c)

where (x - a)² represents the other double root, or point of tangency. After fumbling around a bit, it can be shown that
a = 1/(3)^(1/4), and that k = (2/(3^(3/4)).

Answer 2

Yeh, just considering circles with their center on the x-axis, a circle of any radius fulfills the condition. These circles are described by the function (x - r)² + y² = r², which can be rewritten as x² + y² = 2rx. Substituting y = x³, we get r = (x^6 + x²)/2x = (x^5 + x)/2. The (x^5 + x)/2 is the other point of intersection, and it is clear every value of x gives a different r.

Any chance by "touching" you mean tangent?

Answer 3

sqrt 2 / 2
The first solution that comes to mind is that the circle contains the points (0,0) and (1,1) So its diameter is sqrt 2 (that is, the distance from (0,0) to (1,1) and its radius is half that
However, I'm not at all certain that this is a unique solution. In fact, if you picture the curve y = x^3 and a family of circles having one point on the origin and one point on the curve y = x^3, there is conceivably an infinite number of such circles. Each would be a solution to the system
y = x^3 and sqrt (r^2 - x^2) = y
Since r is undetermined, this system would have an infinite number of solutions.