The ordinates of the points P and Q on the pa | vedclass

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(A) The parabola has focus $(3,0)$ and directrix $x = -3$. The vertex is $(0,0)$ and $a = 3$. The equation of the parabola is $y^2 = 4ax = 12x$.Let th

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$16$

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(A) The parabola has focus $(3,0)$ and directrix $x = -3$. The vertex is $(0,0)$ and $a = 3$. The equation of the parabola is $y^2 = 4ax = 12x$.Let the points be $P(3t_1^2, 6t_1)$ and $Q(3t_2^2, 6t_2)$.The ordinates are in the ratio $3:1$,so $6t_1 / 6t_2 = 3/1$,which implies $t_1 = 3t_2$.The point of intersection $R(\alpha, \beta)$ of tangents at $P$ and $Q$ is given by $\alpha = at_1t_2 = 3(3t_2)(t_2) = 9t_2^2$ and $\beta = a(t_1 + t_2) = 3(3t_2 + t_2) = 12t_2$.Now,calculate $\frac{\beta^2}{\alpha} = \frac{(12t_2)^2}{9t_2^2} = \frac{144t_2^2}{9t_2^2} = 16$.

The ordinates of the points $P$ and $Q$ on the parabola with focus $(3,0)$ and directrix $x = -3$ are in the ratio $3:1$. If $R(\alpha, \beta)$ is the point of intersection of the tangents to the parabola at $P$ and $Q$,then $\frac{\beta^2}{\alpha}$ is equal to $.............$.

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