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The number of integral points on the circle, touching the parabola y 2 = 8 x at 2 , 4 and passing through 0 , 4 , are equal to

Options

  1. A1
  2. B2
  3. C4
  4. D3

Correct answer

C. 4

Step-by-step solution

Equation of tangent at 2,4 on the parabola y 2 = 8 x is y 4 = 8 x + 2 2 ⇒ y = x + 2 Let the equation of the circle touching line y = x + 2 at 2,4 is x - 2 2 + y - 4 2 + λ x - y + 2 = 0 which passes through 0 , 4 ⇒ 4 + 0 + λ 0 - 4 + 2 ⇒ λ = 2 ⇒ Required circle is x 2 + y 2 - 2 x - 10 y + 24 = 0 ⇒ x - 1 2 + y - 5 2 = 2 If x and y are integers, then x - 1 2 = 1 = y - 5 2 ⇒ x = 0 , 2 and y = 4 , 6 ⇒ 4 integral points lie on the circle

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