If the tangents and normals at the extremitie | vedclass

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(C) Let the extremities of the focal chord be $P(at_1^2, 2at_1)$ and $Q(at_2^2, 2at_2)$. Since $PQ$ is a focal chord,$t_1t_2 = -1$.The intersection of

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$y_1 = y_2$

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(C) Let the extremities of the focal chord be $P(at_1^2, 2at_1)$ and $Q(at_2^2, 2at_2)$. Since $PQ$ is a focal chord,$t_1t_2 = -1$.The intersection of tangents at $P$ and $Q$ is $(x_1, y_1) = (at_1t_2, a(t_1 + t_2))$.Substituting $t_1t_2 = -1$,we get $x_1 = -a$ and $y_1 = a(t_1 + t_2)$.The intersection of normals at $P$ and $Q$ is $(x_2, y_2) = (a(t_1^2 + t_2^2 + t_1t_2 + 2), -at_1t_2(t_1 + t_2))$.Substituting $t_1t_2 = -1$,we get $x_2 = a(t_1^2 + t_2^2 + 1)$ and $y_2 = a(t_1 + t_2)$.Comparing the coordinates,we see that $y_1 = y_2$.

If the tangents and normals at the extremities of a focal chord of a parabola $y^2 = 4ax$ intersect at $(x_1, y_1)$ and $(x_2, y_2)$ respectively,then:

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