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Let A lies on 3 x - 4 y + 1 = 0, B lies on 4 x + 3 y - 7 = 0 and C is - 2 , 5 . If A B C D is a rhombus, then the locus of D is a conic whose length of the latus rectum is equal to

Options

  1. A1 0 units
  2. B15 units
  3. C5 units
  4. D20 units

Correct answer

A. 1 0 units

Step-by-step solution

Let, D h , k C - 2 , 5 lies on 4 x + 3 y - 7 = 0 ⇒ B C is perpendicular to 3 x - 4 y + 1 = 0 ⇒ D A is perpendicular to 3 x - 4 y + 1 = 0 ⇒ Because D A = D C so, the distance of D from 3 x - 4 y + 1 = 0 is equal to D C ⇒ h + 2 2 + k - 5 2 = 3 h - 4 k + 1 5 ⇒ 25 x + 2 2 + y - 5 2 = 3 x - 4 y + 1 2 Which represents parabola with focus - 2 , 5 and directrix is 3 x - 4 y + 1 = 0 ⇒ Length of the latus rectum is 2 - 6 - 20 + 1 5 = 10 units

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