If the normals at two points on the parabola | vedclass

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(B) Let the two points on the parabola $y^2 = 4ax$ be $P(at_1^2, 2at_1)$ and $Q(at_2^2, 2at_2)$.The equation of the normal at point $t$ is $y = -tx +

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$8a^2$

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(B) Let the two points on the parabola $y^2 = 4ax$ be $P(at_1^2, 2at_1)$ and $Q(at_2^2, 2at_2)$.The equation of the normal at point $t$ is $y = -tx + 2at + at^3$.If the normals at $t_1$ and $t_2$ meet at a point $(x, y)$ on the parabola,then the point of intersection of the normals is given by $x = 2a + a(t_1^2 + t_1t_2 + t_2^2)$ and $y = -at_1t_2(t_1 + t_2)$.Since the intersection point lies on the parabola $y^2 = 4ax$,we have $(-at_1t_2(t_1 + t_2))^2 = 4a(2a + a(t_1^2 + t_1t_2 + t_2^2))$.This simplifies to $a^2t_1^2t_2^2(t_1 + t_2)^2 = 4a^2(2 + t_1^2 + t_1t_2 + t_2^2)$.For the normals to meet on the parabola,the condition is $t_1t_2 = 2$.The ordinates of the points are $y_1 = 2at_1$ and $y_2 = 2at_2$.The product of the ordinates is $y_1y_2 = (2at_1)(2at_2) = 4a^2(t_1t_2)$.Substituting $t_1t_2 = 2$,we get $y_1y_2 = 4a^2(2) = 8a^2$.

If the normals at two points on the parabola $y^2 = 4ax$ meet on the parabola,then what is the product of the ordinates of these two points?

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