If the circles $x^2 + y^2 + (3 + \sin \beta)x + (2\cos \alpha)y = 0$ and $x^2 + y^2 + (2\cos \alpha)x + 2cy = 0$ touch each other, then the maximum value of $c$ is ___.
Step-by-Step Solution
Key Concept: Two circles through the origin have the same tangent at the origin if and only if their tangent line equations are identical.
Given two circles passing through the origin with equations $x^2 + y^2 + (3 + \sin\beta)x + (2\cos\alpha)y = 0$ and $x^2 + y^2 + (2\cos\alpha)x + 2cy = 0$, their tangents at the origin must be identical. Comparing the tangent equations $(3 + \sin\beta)x + (2\cos\alpha)y = 0$ and $(2\cos\alpha)x + 2cy = 0$, we equate coefficients to get $\frac{3 + \sin\beta}{2\cos\alpha} = \frac{2\cos\alpha}{2c}$. This yields $c = \frac{2\cos^2\alpha}{3 + \sin\beta}$ or $c_{max} = 1$ when $\sin\beta = -1$ and $\alpha = 0$.
Correct Answer: 1