If a normal chord of the parabola y^2 = 4ax s | vedclass

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(B) Let $P(at_1^2, 2at_1)$ be one endpoint of the normal chord,then the other endpoint is $Q(at_2^2, 2at_2)$.For a normal chord,the relation between p

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$\pm \sqrt{2}$

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(B) Let $P(at_1^2, 2at_1)$ be one endpoint of the normal chord,then the other endpoint is $Q(at_2^2, 2at_2)$.For a normal chord,the relation between parameters is $t_2 = -t_1 - \frac{2}{t_1} \dots (1)$.The slope of $OP$ is $m_1 = \frac{2at_1}{at_1^2} = \frac{2}{t_1}$.The slope of $OQ$ is $m_2 = \frac{2at_2}{at_2^2} = \frac{2}{t_2}$.Since the chord subtends a right angle at the vertex $O(0,0)$,the product of slopes is $-1$:$m_1 	imes m_2 = -1$ $\Rightarrow \frac{2}{t_1} 	imes \frac{2}{t_2} = -1$ $\Rightarrow t_1t_2 = -4 \dots (2)$.Substituting $(2)$ into $(1)$:$-4/t_1 = -t_1 - 2/t_1$ $\Rightarrow -4/t_1 = -(t_1^2 + 2)/t_1$ $\Rightarrow 4 = t_1^2 + 2$ $\Rightarrow t_1^2 = 2$ $\Rightarrow t_1 = \pm \sqrt{2}$.The slope of the line joining the vertex and the endpoint $P$ is $m_1 = \frac{2}{t_1} = \frac{2}{\pm \sqrt{2}} = \pm \sqrt{2}$.

If a normal chord of the parabola $y^2 = 4ax$ subtends a right angle at the vertex,find the slope of the line joining the vertex and the endpoint of the normal.

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