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Suppose the line \( 3 p x+2 \sqrt{1-p^{2}} y=1 \) touches a fixed ellipse \( E \) for all \( p \in[-1,1] \). Then
(a) eccentricity of \( E \) is \( \frac{\sqrt{5}}{3} \)
(b) foci of \( E \) are \( \left(0, \frac{\sqrt{5}}{6}\right) \) and \( \left(0, \frac{-\sqrt{5}}{6}\right) \)
(c) equation of a directrix of \( E \) is \( y=\frac{3 \sqrt{5}}{10} \)
(d) equation of a director circle is \( 36\left(x^{2}+y^{2}\right)=13 \)
\( \mathrm{P} \)
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