A circle of radius r touching both sides of a parabola having focal length a. A line segment tangent to the circle is split in two equal segments of length d. What is d in terms of r and a?

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## Solution

The segment length d=r+2a.

One can place the origin of a set of Cartesian coordinates in the parabola vertex and the y axis being it’s axis of symmetry. The parabola is then parametrised by y=x^{2}/4a.

Naming the x-coordinate of the tangency point b, we can find the distances as shown above. We used the right triangle with constant height 2a, shown below.

Applying the Pythagorean theorem in the triangle gives b^{2}=r^{2}-4a^{2}. Now equating the vertical sides of the rectangle gives d^{2}/4a=b^{2}/4a+2a+r. Eliminating b from these two equations gives d=r+2a.

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