jmc

Question

The circles C_1 and C_2 are defined by the equations x^2 + y^2 = 1 and (x - 2)^2 + y^2 = 16, respectively. Find the locus of the centers (a,b) of all circles externally tangent to C_1 and internally tangent to C_2. Enter your answer in the form Pa^2 + Qb^2 + Ra + Sb + T = 0,where all the coefficients are integers, P is positive, and (|P|,|Q|,|R|,|S|,|T|) = 1. Note: The word "locus" is a fancy word for "set" in geometry, so "the locus of the centers" means "the set of the centers".

Accepted Answer

Let (a,b) be the center of a circle that is tangent to C_1 and C_2, and let r be the radius. FIGURE Then the square of the distance of the center of this circle from the center of C_1 is a^2 + b^2 = (r + 1)^2 and the square of the distance of the center of this circle from the center of C_2 is (a - 2)^2 + b^2 = (4 - r)^2. Subtracting these equations, we get a^2 - (a - 2)^2 = (r + 1)^2 - (4 - r)^2.This simplifies to 4a - 4 = 10r - 15, so r = 4a + 1110. Substituting into the equation a^2 + b^2 = (r + 1)^2, we get a^2 + b^2 = ( 4a + 2110 )^2.This simplifies to 84a^2 + 100b^2 - 168a - 441 = 0.