The point of intersection of the tangents of the parabola $y^{2} = 4 x$ , drawn at end points of the chord $x + y = 2$ lies on

x - 2 y = 0

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x + 2 y = 0

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y - x = 0

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x + y = 0

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Solution

The correct option is D $y - x = 0$
Let point of intersection be $(α, β)$
Therefore, chord of contact w.r.t. this point is
$β y = 2 x + 2 α$
$\Rightarrow 2 x - β y = - 2 α$
which is same as $x + y = 2$ .
Therefore comparing, we get $α = β = - 2$
which satisfies $y - x = 0$

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