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JEE Main202126 Aug 2021Morning ShiftMathematicsParabolaActual

If a line along a chord of the circle 4 x 2 + 4 y 2 + 120 x + 675 = 0 , passes through the point ( - 30 , 0 ) and is tangent to the parabola y 2 = 30 x , then the length of this chord is:

Options

  1. A5
  2. B7
  3. C3 5
  4. D5 3

Correct answer

C. 3 5

Step-by-step solution

We have, 4 x 2 + 4 y 2 + 120 x + 675 = 0 ⇒ x 2 + y 2 + 30 x + 675 4 = 0 Centre ≡ - 15 , 0 and radius = 225 - 675 4 = 15 2   units Let the chord cuts the circle at A   &   B and passes through P - 30 , 0 Equation of tangent to parabola y 2 = 30 x is y = m x + 30 4 m       . . . i It passes through - 30 , 0 , so 0 = - 30 m + 30 4 m ⇒ 4 m 2 = 1 ⇒ m = ± 1 2 For m = 1 2 , we have y = x 2 + 15 ⇒ x - 2 y + 30 = 0 Length of O B = - 15 + 0 + 30 5 = 3 5 Len

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