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Match the following.
Column–I Column–II
A The normal chord at a point $(t^{2}, 2 t)$ on the parabola $y^{2} = 4 x$ subtends a right angle at the vertex, then $t^{2}$ is P 4
B The area of the triangle inscribed in the curve $y^{2} = 4 x,$ whose vertices are (1, 2), (4, 4), (16, 8) is Q 2
C The number of distinct normals possible from $(\frac{11}{4}, \frac{1}{4})$ to the parabola $y^{2} = 4 x$ is R 3
D The normal at (a, 2a) on $y^{2} = 4 a x$ meets the curve again at $(a t^{2}, 2 a t),$ then the value of $| t - 1 |$ is S 6

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$A - Q, B - S, C - R, D - S$

$A - P, B - S, C - Q, D - R$

$A - Q, B - S, C - Q, D - P$

$A - R, B - S, C - Q, D - S$

answer is C.

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Detailed Solution

$(a) t^{2} = 2$

$(b) a^{2} | (t_{1} - t_{2}) (t_{2} - t_{3}) (t_{3} - t_{1}) |$

$1^{2} \times 2 \times 3 = 6$

$(∵ t_{1} = 1, t_{2} = 2, t_{3} = 4)$

$y + t x = 2 t + t^{3}$

$\frac{1}{4} + \frac{11}{4} t = 2 t + t^{3}$

$4 t^{3} - 3 t - 1 = 0$

$(t - 1) {(2 t + 1)}^{2} = 0$

2 distinct normals are drawn through $(\frac{11}{4}, \frac{1}{4})$

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