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JEE Advanced2021MathematicsParabolaActual

Let E denote the parabola y 2 = 8 x . Let P = - 2 , 4 , and let Q and Q ' be two distinct points on E such that the lines P Q and P Q ' are tangents to E . Let F be the focus of E . Then which of the following statements is (are) TRUE?

Options

  1. AThe triangle P F Q is a right-angled triangle
  2. BThe triangle Q P Q ' is a right-angled triangle
  3. CThe distance between P and F is 5 2
  4. DF lies on the line joining Q and Q '

Correct answer

A. The triangle P F Q is a right-angled triangle

Step-by-step solution

y 2 = 8 x ⇒ a = 2 , Focus F 2 , 0 , and Directrix   x = − 2 ∵   P − 2 , 4 lies on directrix ⇒   Q Q ' will be a focal chord and tangents are mutually perpendicular. ⇒     Δ Q P Q ' is right angled. Also the portion between point of contact and directrix of parabola subtends right angle at the focus. So, △ P F Q is a right- angled triangle. P F = − 2 − 2 2 + 4 − 0 2 = 4 2

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