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NTA Abhyas JEE Main2020MathematicsParabolaPractice

The tangent to the parabola y = x 2 - 2 x + 8 at P 2 , 8 touches the circle x 2 + y 2 + 18 x + 14 y + λ = 0 at Q . The coordinates of point Q are

Options

  1. A- 7 , - 12
  2. B- 9 , - 13
  3. C- 11 , - 16
  4. D- 31 5 , - 42 5

Correct answer

D. - 31 5 , - 42 5

Step-by-step solution

The equation of the tangent at P 2 , 8 to the parabola is y + 8 2 = 2 x - x + 2 + 8 ⇒ 2 x - y + 4 = 0 . . . ( i ) So, the slope of the normal to the circle is - 1 2 The equation of normal to the circle at Q will be y + 7 = - 1 2 x + 9 x + 2 y + 23 = 0 . . . ( ii ) The point Q is the intersection of the tangent at P and the normal at Q On solving the equation ( i ) & ( ii ) , we get, x + 2 2 x + 4 + 23 = 0 ⇒ 5 x = - 31 ⇒ x = - 31 5 and y = 2 - 31 5 + 4 = - 42 5 Hence, the coordinates of the point Q are - 31 5 , - 42

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