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Let x=2 t, y=\frac{t^{2}}{3} be a conic. Let S be the focus and B be the point on the axis of the conic such that SA \perpBA, where A is any point on the conic. If k is the ordinate of the centroid of the \Delta S A B \text {, then } \lim _{t \rightarrow 1} k \text { is equal to : }

Option: 1

\frac{17}{18}


Option: 2

\frac{19}{18}


Option: 3

\frac{11}{18}


Option: 4

\frac{13}{18}


Answers (1)

best_answer


\mathrm{Parabola\,\, x^{2}= 12y}

\mathrm{SA\perp SB}
\mathrm{So,\: m_{AS}\cdot m_{AB}= -1}
\mathrm{\frac{\left ( 3-\frac{t^{2}}{3} \right )\left ( \alpha -\frac{t^{2}}{3} \right )}{\left ( 0-2t \right )}\cdot \frac{\left ( \alpha -\frac{t^{2}}{3} \right )}{\left ( 0-2t \right )}= -1}
by solving
\mathrm{3\alpha = \frac{27t^{2}+t^{4}}{t^{2}-9}}
\mathrm{\text{ordinate of centroid of }\triangle SAB= k= \frac{\alpha +\frac{t^{2}}{3}+3}{3}}
\mathrm{k= \frac{9+3\alpha +t^{2}}{9}}
\mathrm{\lim_{t\rightarrow 1}k=\lim_{t\rightarrow } \frac{1}{9}\left ( 9+t^{2} +\frac{27\, t^{2}+t^{4}}{\left ( t^{2}-9 \right )}\right )= \left ( \frac{13}{18} \right )}

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Rakesh

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