Question

In polar coordinates, the parametric equations x=6+cosθ and y=sinθ represent a circle Γ1. In Cartesian coordinates, there is a circle Γ2 that is externally tangent to Γ1, tangent to the y-axis, and centered at (12,sqrt(a)). What is the value of a?

Answer 1

well, we have

Γ1 = (x-6)^2 + y^2 = 1
Γ2 = (x-12)^2 + ay^2 = 144

So, solve to find a. The circles must intersect in only one point.

Answer 2

Thank you.

Answer 3

Nah, too much algebra.

Consider Γ2. Since it has its center at x=12, and touches the y-axis, its radius is 12.

Consider the line joining the centers of the circles. It goes from (6,0) to (12,√a). So, its length is √(a+36).

But, we know the circles are of radius 1 and 12, so

√(a+36) = 13
a+36 = 169
a = √133