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Point where the tangents at two points of a parabola meet

This type of question, asked 3 times: 25 Jul 2022, 28 Jul 2022, 8 Apr 2023. All 3 questions below, oldest first.

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  1. Question 1

    25 Jul 2022

    163
    The tangents at the point A(1,3)A(1,3) and B(1,−1)B(1, - 1) on the parabola y2−2x−2y=1y^{2}- 2x - 2y = 1 meet at the point PP. Then the area (in unit  2\ ^{2} ) of the triangle PABPAB is:-

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  2. Question 2

    28 Jul 2022

    71
    If the tangents drawn at the points PP and QQ on the parabola y2=2x−3y^{2}= 2x - 3 intersect at the point R(0,1)R(0,1), then the orthocentre of the triangle PQRPQR is:

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  3. Question 3

    8 Apr 2023

    176
    The ordinates of the points PP and QQ on the parabola with focus (3,0)(3,0) and directrix x=−3x=-3 are in the ratio 3:13:1. If the tangents to the parabola at PP and QQ meet at the point (α,β)(\alpha,\beta), then β2α\frac{\beta^{2}}{\alpha} is equal to