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Match the following:
Column-I Column-II
A If two distinct chord of a parabola $y^{2} = 4 a x$ passing through the point $(a, 2 a)$ are bisected by the line $x + y = 1$ , then the length of the latus rectum can be p $- 1$
B The parabola $y = x^{2} - 5 x + 4$ cuts the x-axis at P and Q. A circle is drawn through P and Q so that the origin lies outside it. The length of tangent to the circle from the origin is equal to q 0
C If $y + b = m_{1} (x + a)$ and $y + b = m_{2} (x + a)$ are two tangents to $y^{2} = 4 a x$ , then $m_{1} m_{2}$ is equal to r 1
D If the point (h,-1) is exterior to both the parabolas $y^{2} = | x |$ , then the integral part of h can be equal to s 2

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$A \to s; B \to p; C \to q; D \to r$

$A \to s; B \to r, s; C \to p; D \to q$

$A \to s; B \to r, s; C \to q; D \to p$

$A \to r, s; B \to s; C \to p; D \to q$

answer is A.

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Detailed Solution

A)Any point is $(t, 1 - t)$
The chord with this as mid point
$y (1 - t) - 2 a (x + t) = {(1 - t)}^{2} - 4 a t$
$\Rightarrow {(1 - t)}^{2} = 2 a (1 - a) \geq 0 \Rightarrow 0 < a \leq 1$
$∴$ Length of latus Rectum $\in (0, 4]$
B) $P \equiv (1, 0), Q \equiv (4, 0)$
$∴$ Equation of circle passes through two points is $(x - 1) (x - 4) + y^{2} + λ y = 0$
$∴$ Length of the tangent from (0,0) is $\sqrt{4}$ i.e.,2
C) $m_{1}, m_{2} = - 1$
D) $1 - | h | > 0 \Rightarrow - 1 < h < 1$

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