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JEE Advanced 2022 Paper 1 · Mathematics · Conic Sections · hard

Let P and Q be the end points of the focal chord of the parabola y^2=8x with P=(8,8). Let the tangents at P and Q meet at M.

One or more options correct

Let PP and QQ be the end points of the focal chord of the parabola y2=8xy^2=8x with P=(8,8)P=(8,8). Let the tangents at PP and QQ meet at MM. Which of the following statements is/are correct?
  1. A. PQ=12PQ=12.
  2. B. M=(−2,−3)M=(-2,-3).
  3. C. The area of triangle PQMPQM is 1254\dfrac{125}{4}.
  4. D. The normal at PP meets the parabola again at (18,12)(18,12).
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Answer: C

With parametrisation (2t2,4t)(2t^2,4t), PP is t=2t=2; for a focal chord t1t2=−1t_1t_2=-1, so QQ is t=−12t=-\frac12, i.e. (12,−2)\left(\frac12,-2\right). (A) PQ=x1+x2+4=8+12+4=252≠12PQ=x_1+x_2+4=8+\frac12+4=\frac{25}2\ne12: false. (B) Tangents ty=x+2t2ty=x+2t^2 at t=2,−12t=2,-\frac12 meet at (2t1t2,2(t1+t2))=(−2,3)(2t_1t_2,2(t_1+t_2))=(-2,3), not (−2,−3)(-2,-3): false. (C) Area with vertices (8,8),(12,−2),(−2,3)(8,8),(\frac12,-2),(-2,3): 12∣8(−2−3)+12(3−8)+(−2)(8+2)∣=12∣−40−52−20∣=1254\frac12|8(-2-3)+\frac12(3-8)+(-2)(8+2)|=\frac12|-40-\frac52-20|=\frac{125}4: true. (D) The normal at tt meets again at t′=−t−2t=−3t'=-t-\frac2t=-3, i.e. (18,−12)(18,-12), not (18,12)(18,12): false.
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