Let \( x=2 t, y=\frac{t^{2}}{3} \) be a conic. Let \( \mathrm{S} \) be the focus and \( B \) be ...

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Let \( x=2 t, y=\frac{t^{2}}{3} \) be a conic. Let \( \mathrm{S} \) be the focus and \( B \) be the point on the axis of the conic such that \( S A \perp B A \), where \( A \) is any point on the conic. If \( k \) is the ordinate of the centroid of \( \triangle S A B \), then \( \lim _{t \rightarrow 1} k \) is equal to
(a) \( \frac{17}{18} \)
(b) \( \frac{19}{18} \)
(c) \( \frac{11}{18} \)
(d) \( \frac{13}{18} \)
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