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The locus of mid-points of all chords of parabola y 2 = 4 x , for which all circles drawn taking them as diameters passes through the vertex of the parabola is a conic whose length of the smallest focal chord is equal to

Options

  1. A1 units
  2. B2 units
  3. C3 units
  4. D4 units

Correct answer

B. 2 units

Step-by-step solution

Circle drawn by taking A B as diameter passes through vertex hence A V is perpendicular to B V Let h , k be the mid-point of the chord ⇒ t 1 t 2 = - 4 ,   h = t 1 2 + t 2 2 2 ,   k = 2 t 1 + t 2 2 k 2 = t 1 2 + t 2 2 + 2 t 1 t 2 ⇒ y 2 = 2 x - 8 Hence, the locus of the conic is y 2 = 2 x - 4 ⇒ Length of the latus rectum (smallest focal chord) = 2 units

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