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.(adsbygoogle = window.adsbygoogle || []).push({});

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).

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# Circle in a parabola

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