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If the line \( y=m x+\sqrt{a^{2} m^{2}-b^{2}}, m=\frac{1}{2} \) touches the
\( \mathrm{P}^{162} \)
W
hyperbola \( \frac{x^{2}}{16}-\frac{y^{2}}{3}=1 \) at the point
\( (4 \sec \theta, \sqrt{3} \tan \theta) \) then \( \theta \) is
(1) \( \frac{\pi}{2} \)
(2) \( \frac{\pi}{4} \)
(3) \( \frac{2 \pi}{3} \)
(4) \( \frac{\pi}{6} \)
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