Find Center of Circle Inscribed in Parabola

[lkh1986]

A circle with the radius of 1 unit is inscribed in a parabola with the equation of y=x^2. Find the coordinate of the center of the circle.

The answer given in the book is (0, 5/4).

I attempt this question by letting the equation of the circle be x^2 + (y-c)^2 = 1, where c is the y-coordinate of the circle. Then, by substitutuing y=x^2 into that equation, I obtain another equation. From the new equation, I try to use the formula b^2 - 4ac = 0. Then, I got the value of c, which is 5/4.

However, what I do not understand is this: The equation b^2 - 4ac = 0 means that 2 function intercept at only 1 point. But in this case, the circle touches the parabola at 2 points.

Do I get the answer c=5/4 by chance? Or is there any other method to solve this problem? Maybe the circle does not intercept the parabola at only 2 points? (I say this because I find that at the points of interception, it seems that the circle and the parabola do not share a same tangent line).

 

[GeoMike]

The resulting equation (when you substitute) is quartic: x^4+(1-2c)x^2+c^2=1
The reason you have two intersection points is because the solution to the quadratic formula in this case is actually x^2

x^2 = (-b +- sqrt(b^2-4ac))/(2a)

Even though the discriminant, being zero, causes the square root to drop out of the quadratic forumla (giving just one value on the right) you still have to take the square root of both sides to solve for x here, which gives two values of x in this case.

-GeoMike-

 

[Architeuthis Dux]

I would first clarify the question and make sure there are no typos or missing information. In this case, I would ask for clarification on the term "inscribed" as it can have different meanings in geometry. Does the circle intersect with the parabola at only two points or does it touch it at two points? This information is crucial in finding the center of the circle.

Assuming that the circle intersects with the parabola at only two points, the method you used to solve the problem is correct. However, it is possible that the given answer is incorrect and the correct answer could be (0, -5/4) as the circle could be inscribed in the parabola from the bottom.

To confirm the correct answer, I would suggest plotting the parabola and the circle on a graph and visually inspecting the points of intersection. Additionally, using calculus, you can find the derivative of the parabola and the circle at the points of intersection and see if they have the same slope, indicating that they share a tangent line.

In conclusion, as a scientist, I would approach this problem by clarifying any ambiguities in the question and using various methods to confirm the correct answer.