The sum of the reciprocals of the focal dista | vedclass

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(A) Let the coordinates of points $P$ and $Q$ on the parabola $y^{2} = 4ax$ be $(at_{1}^{2}, 2at_{1})$ and $(at_{2}^{2}, 2at_{2})$ respectively.Since

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$\frac{1}{a}$

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(A) Let the coordinates of points $P$ and $Q$ on the parabola $y^{2} = 4ax$ be $(at_{1}^{2}, 2at_{1})$ and $(at_{2}^{2}, 2at_{2})$ respectively.Since $PQ$ is a focal chord,$t_{1}t_{2} = -1$.The focal distances of points $P$ and $Q$ are $r_{1} = a(1 + t_{1}^{2})$ and $r_{2} = a(1 + t_{2}^{2})$.The sum of the reciprocals is $\frac{1}{r_{1}} + \frac{1}{r_{2}} = \frac{1}{a(1 + t_{1}^{2})} + \frac{1}{a(1 + t_{2}^{2})}$.Substituting $t_{2} = -\frac{1}{t_{1}}$,we get $\frac{1}{a(1 + t_{1}^{2})} + \frac{1}{a(1 + \frac{1}{t_{1}^{2}})} = \frac{1}{a(1 + t_{1}^{2})} + \frac{t_{1}^{2}}{a(t_{1}^{2} + 1)}$.$= \frac{1 + t_{1}^{2}}{a(1 + t_{1}^{2})} = \frac{1}{a}$.

The sum of the reciprocals of the focal distances of a focal chord $PQ$ of the parabola $y^{2} = 4ax$ is:

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