Example 1 Given point $Q(2,0)$ and circle $C: x^{2}+y^{2}=1$, the ratio of the length of the tangent from a moving point $M$ to circle $C$ to $|M Q|$ is a constant $\lambda(\lambda>0)$, find the equation of the locus of the moving point $M$.

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Let $M(x, y)$, and the tangent line from $M$ to the circle be $M N$. Then $|M N|=\sqrt{|M O|^{2}-|O N|^{2}}=$ $\sqrt{x^{2}+y^{2}-1}$. Also, $|M Q|=\sqrt{(x-2)^{2}+y^{2}}$. According to the problem, we have $\frac{M N}{|M Q|}=\lambda$, which means
$$
\frac{\sqrt{x^{2}+y^{2}-1}}{\sqrt{(x-2)^{2}+y^{2}}}=\lambda,
$$

Simplifying, we get $\left(\lambda^{2}-1\right) x^{2}+\left(\lambda^{2}-1\right) y^{2}-4 \lambda^{2} x+4 \lambda^{2}+1=0$.
When $\lambda=1$, the equation is $x=\frac{5}{4}$, representing a straight line; when $\lambda \neq 1$, the equation represents a circle with center $\left(\frac{2 \lambda^{2}}{\lambda^{2}-1}, 0\right)$ and radius $\frac{\sqrt{3 \lambda^{2}+1}}{1 \lambda^{2}-1}$.