Extrema

[L.E.]

Consider a and b fixed, with b<a. Let c be a variable such that b<c<a. Let (phi) be the acute angle between the tangents to the circle x^2 + y^2 = c^2 and the ellipse (bx)^2 + (ay)^2 = (ab)^2 at a point of intersection. Find tan(phi) when c is chosen so that (phi) is greatest.

My approach:

1) Find points of intersection and choose one of them.

2) Write the circle and ellipse as functions defined at the chosen point of intersection.

3) Differentiate the functions and evaluate the derivitives at the point of intersection.

4) Write phi = arctan(dy(circle)/dx) - arctan(dy(ellipse)/dx)

5) Find c such that d(phi)/dc = 0

6) Evaluate phi at this value of c

7) Evaluate tan(phi)

This approach worked, but took many pages of work. Does anyone know of a more clever approach?

[piero]

i cannot see another approach fundamentally different than yours.
the steps below are a different spin on your approach that might (or might not) minimize the computations.

(1) at the intersection point (x_0, y_0) in 1st quadrant between circle and ellipse, the tangent vector to the circle is (-y_0, x_0) and the tangent vector to the ellipse is (-a*y_0/b, b*x_0/a).

(2) using the fact that (x_0, y_0) belongs to both the circle and the ellipse, compute the dot product of the two vectors above.
if E = (a*y_0/b)^2 + (b*x_0/a)^2, i got sin(phi) = a*b/ square_root(E).

(3) you want to maximize phi, so this means you want to minimize E. write x_0 and y_0 in terms of only a, b, and c, plug them in E, who now becomes a function of a, b, and c.

(4) solve dE/dc=0 for c (such c must be unique), find the corresponding E, and then compute sin(phi) and tan(phi).

just curious why you want to maximize phi.