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The locus of the mid points of all chords of the parabola y2 = 4x, which are drawn through its vertex, is
1. y2 = 8x
2. y2 = 2x
3. x2 + 4y2 = 16
4. x2 = 2y

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Correct Answer - Option 2 : y2 = 2x

Concept:

Let the mid point of chord of the parabola y2 = 4ax is (h, k)

The equation of chord passing through the mid point of the chord of the parabola is given by T = S1 i.e yk - 2a(x + h) = k2 - 4ah, where T is the tangent and S1 is equation of the parabola which we get by replacing y and x by k and h respectively.

Calculation:

compare the parabola  y2 = 4x with parabola y2 = 4ax, we get a = 1

chord passes through vertex (0, 0) so this point satisfy chord equation

⇒ 0 - 2(0 + h) = k2 - 4h

⇒ k2 = 2h

Now replace h by x and k by y in the above equation, we get

y2 = 2x

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