Problem: Consider (1) the parabola y=3-1/10 x^2 and (2) the upper half of the circle centered at (20, 0) with radius of 10. Find the points on the parabola where the tangent line is also tangent to the upper half of the circle. (You can find these points in exact form in Mathematica, but they are so messy that you will want to approximate them.) Make a single, final graph of the parabola, semicircle, and tangent line(s).

Question

Problem:  Consider (1) the parabola y=3-1/10 x^2  and (2) the upper half of the circle centered at (20, 0) with radius of 10.  Find the points on the parabola where the tangent line is also tangent to the upper half of the circle.  (You can find these points in exact form in Mathematica, but they are so messy that you will want to approximate them.)  Make a single, final graph of the parabola, semicircle, and tangent line(s).

Hint to plot a circle that really looks like a circle:  In Show, set PlotRange->{{xmin, xmax},{ymin,ymax}} with the distance between xmin and xmax the same as the distance from ymin to ymax.  Also set AspectRatio -> 1.

Steve · Answer

at x=h, the tangent line to the parabola is

y - (3-h^2/10) = (-h/5)(x-h)
y = -h/5 x + 3+h^2/10

So, find where the line intersects the circle, and find h so that there is only one solution.

It might be useful to pick a value of x close to what you expect, and then use an iterative method to close in on the needed slope for the line.

http://www.wolframalpha.com/input/?i=plot+y%3D3-x%5E2%2F10,+(x-20)%5E2%2By%5E2+%3D+100,+y%3D0,+x%3D0