Asked by Anonymous
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?
Answers
Answered by
Steve
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.
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.
Answered by
Nicholas
Thank you.
Answered by
Steve
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
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
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