If the tangents at the extremities of a chord | vedclass

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(B) Let the equation of the parabola be $y^{2} = 4ax$ and $P(at_{1}^{2}, 2at_{1}), Q(at_{2}^{2}, 2at_{2})$ be the extremities of the chord $PQ$.The co

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$G.P.$

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(B) Let the equation of the parabola be $y^{2} = 4ax$ and $P(at_{1}^{2}, 2at_{1}), Q(at_{2}^{2}, 2at_{2})$ be the extremities of the chord $PQ$.The coordinates of $T$,the point of intersection of the tangents at $P$ and $Q$,are $(at_{1}t_{2}, a(t_{1} + t_{2}))$.The distance of the focus $S(a, 0)$ from $P$ is $SP = \sqrt{(at_{1}^{2} - a)^{2} + (2at_{1} - 0)^{2}} = a(1 + t_{1}^{2})$.Similarly,the distance of the focus $S$ from $Q$ is $SQ = a(1 + t_{2}^{2})$.The distance of the focus $S$ from $T$ is $ST = \sqrt{(at_{1}t_{2} - a)^{2} + (a(t_{1} + t_{2}) - 0)^{2}}$.$ST^{2} = a^{2}(t_{1}^{2}t_{2}^{2} - 2t_{1}t_{2} + 1 + t_{1}^{2} + 2t_{1}t_{2} + t_{2}^{2}) = a^{2}(1 + t_{1}^{2} + t_{2}^{2} + t_{1}^{2}t_{2}^{2}) = a^{2}(1 + t_{1}^{2})(1 + t_{2}^{2})$.Thus,$ST^{2} = SP \cdot SQ$,which implies that $SP, ST, SQ$ are in $G.P.$

List-$A$	List-$B$
$(A)$. The vertex of the parabola $y^2+4x-2y+3=0$ is	$(I)$. $\left(\frac{5}{4}, 1\right)$
$(B)$. The vertex of the parabola $x^2+8x+12y+4=0$ is	$(II)$. $\left(1, \frac{5}{4}\right)$
$(C)$. The focus of the parabola $y^2-x-2y+2=0$ is	$(III)$. $\left(-\frac{1}{2}, 1\right)$
$(D)$. The focus of the parabola $x^2-2x-8y-23=0$ is	$(IV)$. $(1, -1)$
	$(V)$. $(-4, 1)$

If the tangents at the extremities of a chord $PQ$ of a parabola intersect at $T$,then the distances of the focus of the parabola from the points $P, T, Q$ are in

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