Consider the ellipse \( \frac{x^{2}}{4}+\frac{y^{2}}{3}=1 \). Let \( H(\alpha, 0), 0\alpha2 \), ...

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Consider the ellipse \( \frac{x^{2}}{4}+\frac{y^{2}}{3}=1 \). Let \( H(\alpha, 0), 0\alpha2 \), be a point. A straight line drawn through \( H \) parallel to \( y \)-axis crosses the ellipse and its audiliary circle at points \( E \) and \( F \) respectively, in the first quadrant. The tangents to the ellipse at the point \( E \) intersects the positive \( x \)-axis at a point \( G \). Suppose the straight line joining \( F \) and the origin makes an angle \( \phi \) with the positive \( x \)-axis.
(i) If \( \phi=\frac{\pi}{4} \), then find the area of the triangle \( F G H \)
(ii) If \( \phi=\frac{\pi}{12} \), then find the area of the triangle \( F G H \)
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