For what value of a are the circles ax^2 + a^2y^2 = 9 and x^2 + y^2 = 9 tangential to each other.

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If the circles ax^2 + a^2y^2 = 9 and x^2 + y^2 = 9 are tangential to each other, they touch each other at a single point. Let it be (X, Y)

We have two equations:

aX^2 + a^2Y^2 = 9 ...(1)

X^2 + Y^2 = 9 ...(2)

(1) - a*(2)

=> aX^2 + a^2Y^2 - aX^2 - aY^2 = 9 - 9a

=> Y^2(a^2 - a) = 9(1 - a)

=> Y^2a(a - 1) = 9(1 - a)

=> Y^2*a = -9

We have 2 variables Y and a and only one equation. It is not possible to find a unique solution for a.

There can be an infinite number of values of a for which the circles  ax^2 + a^2y^2 = 9 and x^2 + y^2 = 9 are tangential to each other.

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