If the normals at two points P and Q of a par | vedclass

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(D) Let the points $P$ and $Q$ on the parabola $y^2 = 4ax$ be $(at_1^2, 2at_1)$ and $(at_2^2, 2at_2)$ respectively.The equation of the normal at point

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(D) Let the points $P$ and $Q$ on the parabola $y^2 = 4ax$ be $(at_1^2, 2at_1)$ and $(at_2^2, 2at_2)$ respectively.The equation of the normal at point $t$ is $y = -tx + 2at + at^3$.If the normals at $t_1$ and $t_2$ intersect at a point $R(at_3^2, 2at_3)$ on the parabola,then $t_3$ must satisfy the relation $t_1t_2 + t_3(t_1 + t_2 + t_3) + 2 = 0$ is not the standard form,but rather for the intersection of normals at $t_1, t_2$ meeting at $t_3$,the condition is $t_1t_2 = 2$ and $t_3 = -(t_1 + t_2)$.The ordinates of $P$ and $Q$ are $y_1 = 2at_1$ and $y_2 = 2at_2$.The product of the ordinates is $y_1y_2 = (2at_1)(2at_2) = 4a^2(t_1t_2)$.Since $t_1t_2 = 2$,the product is $4a^2(2) = 8a^2$.

If the normals at two points $P$ and $Q$ of a parabola $y^2 = 4ax$ intersect at a third point $R$ on the curve,then the product of the ordinates of $P$ and $Q$ is (in $a^2$)

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