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(B) Let the equation of the parabola be $y^{2} = 4ax$ and the endpoints of the chord $PQ$ be $P(at_1^{2}, 2at_1)$ and $Q(at_2^{2}, 2at_2)$.The point o

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$G.P.$

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(B) Let the equation of the parabola be $y^{2} = 4ax$ and the endpoints of the chord $PQ$ be $P(at_1^{2}, 2at_1)$ and $Q(at_2^{2}, 2at_2)$.The point of intersection $T$ of the tangents at $P$ and $Q$ has coordinates $(at_1t_2, a(t_1 + t_2))$.The distance of a point $(at^{2}, 2at)$ from the focus $S(a, 0)$ is $a(1 + t^{2})$.Thus,$SP = a(1 + t_1^{2})$ and $SQ = a(1 + t_2^{2})$.The distance $ST$ from the focus $S(a, 0)$ to $T(at_1t_2, a(t_1 + t_2))$ is given by:$ST^{2} = (at_1t_2 - a)^{2} + (a(t_1 + t_2) - 0)^{2}$$ST^{2} = a^{2}(t_1^{2}t_2^{2} - 2t_1t_2 + 1 + t_1^{2} + 2t_1t_2 + t_2^{2})$$ST^{2} = a^{2}(t_1^{2}t_2^{2} + t_1^{2} + t_2^{2} + 1)$$ST^{2} = a^{2}(1 + t_1^{2})(1 + t_2^{2})$$ST^{2} = SP \cdot SQ$Since $ST^{2} = SP \cdot SQ$,the distances $SP, ST, SQ$ are in $G.P.$

If the tangents at the endpoints $P$ and $Q$ of a chord of a parabola meet at point $T$,then the distances of points $P, T, Q$ from the focus of the parabola are in which progression?

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