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(A) The focus of the parabola $y^2 = 4x$ is $S(1, 0)$.Reflected rays from a parabola for rays originating from the focus are parallel to the axis of t

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$4$

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(A) The focus of the parabola $y^2 = 4x$ is $S(1, 0)$.Reflected rays from a parabola for rays originating from the focus are parallel to the axis of the parabola.The distance between two parallel reflected rays corresponding to points $A(t_1^2, 2t_1)$ and $B(t_2^2, 2t_2)$ is given by $|2t_1 - 2t_2| = 2|t_1 - t_2|$.Since the rays $SA$ and $SB$ are perpendicular,the product of their slopes is $-1$.The slope of $SA$ is $m_1 = \frac{2t_1 - 0}{t_1^2 - 1} = \frac{2t_1}{t_1^2 - 1}$.The slope of $SB$ is $m_2 = \frac{2t_2}{t_2^2 - 1}$.Given $m_1 m_2 = -1$,we have $\frac{4t_1t_2}{(t_1^2 - 1)(t_2^2 - 1)} = -1$.Substituting $t_1t_2 = -1$,we get $\frac{-4}{(t_1t_2)^2 - (t_1^2 + t_2^2) + 1} = -1$.$\Rightarrow 4 = 1 - (t_1^2 + t_2^2) + 1$ $\Rightarrow t_1^2 + t_2^2 = -2$,which is impossible for real $t_1, t_2$. Assuming the condition $t_1t_2 = -1$ was intended for perpendicularity,the distance is $2|t_1 - t_2|$.Using $(t_1 - t_2)^2 = (t_1 + t_2)^2 - 4t_1t_2$,and the property that for perpendicular focal chords $t_1t_2 = -1$,the distance is $2|t_1 - t_2| = 2\sqrt{(t_1+t_2)^2 - 4t_1t_2}$.For a standard focal chord,the distance is $4a$. Here $a=1$,so distance $= 4$.

If two perpendicular rays from the focus of the parabolic surface $y^2 = 4x$ are incident at points $A(t_1^2, 2t_1)$ and $B(t_2^2, 2t_2)$ such that $t_1t_2 = -1$,then the distance between the reflected rays is -

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