If the chord of contact of the circle $x^{2}+y^{2}-2x+4y+\lambda=0$ with respect to a point lying on the circle $x^{2}+y^{2}-2x+4y+1=0$ touches the circle $x^{2}+y^{2}-2x+4y+3=0$, then the number of value(s) of $\lambda$ is

The center of the circles is (1, -2). Let P be on x^2 + y^2 - 2x + 4y + 1 = 0, meaning its distance squared from the center is 8. The radius of x^2 + y^2 - 2x + 4y + 3 = 0 squared is 4 - lambda. The chord of contact from P to this third circle is a tangent, so using the tangent length formula, the distance squared from P to (1, -2) equals the radius squared plus the tangent length squared, which implies 8 equals 4 - lambda plus 8. This yields lambda = 4, which means the circle x^2 + y^2 - 2x + 4y + lambda = 0 has radius zero and the chord of contact cannot be a tangent. Thus, there are zero valid values for lambda.

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