When does this equation in $x, y$ and $\theta$ represent a circle?

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Given \(\theta\) in the range \(0\leq \theta < \pi\), the equation \[x^2+y^2+4x \cos\theta+8y \sin\theta+10=0\] represents a circle for

1. \(0<\theta<\dfrac{\pi}{3}\);

2. \(\dfrac{\pi}{4}<\theta<\dfrac{3\pi}{4}\);

3. \(0<\theta<\dfrac{\pi}{2}\);

4. all values for \(\theta\).

What is the general form for the equation of a circle? Can we manipulate this equation into that form?