4. Through a point $P$ on the ellipse $y^{2}+\frac{x^{2}}{4}=1$, two tangents are drawn to the circle $x^{2}+y^{2}=1$. The chord of the points of tangency intersects the $x$ and $y$ axes at points $E$ and $F$, respectively. The minimum value of the area of $\triangle E O F$ is $\qquad$.
Given $P(2 \cos \theta, \sin \theta)$, it is easy to know that the line $M N: 2 x \cos \theta+y \sin \theta=1$, then $E\left(\frac{1}{2 \cos \theta}, 0\right), F\left(0, \frac{1}{\sin \theta}\right)$, thus $S_{\triangle O E F}=\frac{1}{2}\left|\frac{1}{\sin 2 \theta}\right| \geqslant \frac{1}{2}$, equality holds when $|\sin 2 \theta|=1 \Rightarrow P\left( \pm \sqrt{2}, \pm \frac{\sqrt{2}}{2}\right)$