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Shortest distance between a parabola and a line or another curve

This type of question, asked 3 times: 2021 Paper 2, 2021 Paper 4, 26 Jul 2022. All 3 questions below, oldest first.

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  1. Question 1

    2021 Paper 2

    66
    If PP is a point on the parabola y=x2+4y=x^{2}+ 4 which is closest to the straight line y=4x−1y= 4x- 1, then the co-ordinates of PP are :

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  2. Question 2

    2021 Paper 4

    77
    The shortest distance between the line x−y=1x-y= 1 and the curve x2=2yx^{2}= 2y is:

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  3. Question 3

    26 Jul 2022

    155
    Let PP and QQ be any points on the curves ( x−x - 1)2+(y+1)2=11)^{2}+ (y + 1)^{2}= 1 and y=x2y =x^{2}, respectively. The distance between PP and QQ is minimum for some value of the abscissa of PP in the interval

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