Assertion :If $P_{1} Q_{1}$ & $P_{2} Q_{2}$ are two focal chords of the parabola $y^{2} = 4 a x$ , then locus of point of intersection of chords $P_{1} P_{2}$ & $Q_{1} Q_{2}$ is directrix of the parabola. Reason: The locus of points of intersection of perpendicular tangents of a parabola is the directrix of parabola.

Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

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Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

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Assertion is correct but Reason is incorrect

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Both Assertion and Reason are incorrect

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Solution

The correct option is B Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
Let $P_{1} \equiv (a t_{1}^{2}, 2 a t_{1}), P_{2} \equiv (a t_{2}^{2}, 2 a t_{2})$ then the points $Q_{1} \equiv (\frac{a}{t_{1}^{2}}, - \frac{2 a}{t_{1}})$ &
$Q_{2} \equiv (\frac{a}{t_{2}^{2}}, - \frac{2 a}{t_{2}})$ Now
$P_{1} Q_{1}$ & $P_{2} Q_{2}$ are
focal chords therefore equation of $P_{1} P_{2}$ is
given by $y (t_{1} + t_{2}) = 2 (x + a t_{1} t_{2}) . . . . . (i)$
replacing $t_{1}$ by $- \frac{1}{t_{1}}$ & $t_{2}$ by $- \frac{1}{t_{2}}$
in (i) we get equation of $Q_{1} Q_{2}$ given by
$y (- \frac{1}{t_{1}} - \frac{1}{t_{2}}) = 2 {x + a (- \frac{1}{t_{1}}) (- \frac{1}{t_{2}})}$ $\Rightarrow - y (t_{1} + t_{2}) = 2 x t_{1} t_{2} + 2 a . . . . (i i)$ From (i) and (ii) we have $2 x t_{1} t_{2} + 2 a + 2 x + 2 a t_{1} t_{2} = 0$ $\Rightarrow 2 (x + a) (1 + t_{1} t_{2}) = 0 \Rightarrow x + a = 0, t_{1} t_{2} + 1 \neq 0$ Which is
equation of directrix of parabola $y^{2} = 4 a x$ Hence
Assertion (A) & Reason (R) both are true but Reason (R) is not the
correct explanation of Assertion (A) Hence choice (b) is correct answer,

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Assertion: All the points (1, 0), (-1, 0), (2, 0) and (5, 0) lie on the x – axis.
Reason: Equation of the x-axis is y = 0

Choose the correct option.

Assertion: If AB = CD, then BE = DE and AE = CE, where E is the point of intersection of AD and BC.

Reason: Angles in the same segment are equal.

Which of the following options is correct?

{2n}^{2}

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