Assertion :The locus of mid points of the chords of the parabola $y^{2} = 4 a x$ which subtends a right angle at the vertex is $y^{2} = 2 a (x - 4 a)$ . Reason: Chord $P Q$ joining the points $t_{1}$ & $t_{2}$ subtends a right angle at the vertex of a parabola if $t_{1} t_{2} = - 2$ .

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 C Assertion is correct but Reason is incorrect
Let P & Q be $t_{1}$ & $t_{2} .$
Since PQ subtend an angle of $90^{\circ}$ at the vertex A so that $\frac{2}{t_{1}} \frac{2}{t_{2}} = - 1$ i.e.
$t_{1} t_{2} = - 4$ so Reason (R) is false. Again let (h, k) be the mid point of PQ, then equation of PQ is $k^{2} - 4 a h = k y - 2 a (x + h)$ or $2 a x - k y + k^{2} - 2 a h = 0$

The combined equation of OP & OQ is $y^{2} = 4 a x (\frac{2 a x - k y}{2 a h - k^{2}})$ or $8 a^{2} x^{2} + (k^{2} - 2 a h) y^{2} - 4 a k x y = 0$
Since $O P ⊥ O Q$ therefore, $8 a^{2} + k^{2} - 2 a h = 0$
Hence locus of (h, k) is $y^{2} - 2 a x + 8 a^{2} = 0 \Rightarrow y^{2} = 2 a (x - 4 a)$ so Assertion is true

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Similar questions

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Assertion :The locus of mid points of the chords of the parabola y2=4ax which subtends a right angle at the vertex is y2=2a(x−4a). Reason: Chord PQ joining the points t1 & t2 subtends a right angle at the vertex of a parabola if t1t2=−2.

Assertion :The locus of mid points of the chords of the parabola $y^{2} = 4 a x$ which subtends a right angle at the vertex is $y^{2} = 2 a (x - 4 a)$ . Reason: Chord $P Q$ joining the points $t_{1}$ & $t_{2}$ subtends a right angle at the vertex of a parabola if $t_{1} t_{2} = - 2$ .