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Ray of light reflected from a line through two given points

This type of question, asked 4 times: 2021 Paper 24, 28 Jun 2022, 8 Apr 2024, 9 Apr 2024. All 4 questions below, oldest first.

Try each one first, then open its answer.

  1. Question 1

    2021 Paper 24

    90
    A man starts walking from the point P(−3,4)P( - 3,4), touches the xx-axis at RR, and then turns to reach at the point Q(0,2)Q(0,2). The man is walking at a constant speed. If the man reaches the point QQ in the minimum time, then 50((PR)2+(RQ)2)50\left( (PR)^{2}+ (RQ)^{2} \right) is equal to

  2. Question 2

    28 Jun 2022

    85
    A ray of light passing through the point P(2,3)P(2,3) reflects on the XX-axis at point AA and the reflected ray passes through the point Q(5,4)Q(5,4). Let RR be the point that divides the line segment AQAQ internally into the ratio 2:12:1. Let the co-ordinates of the foot of the perpendicular MM from RR on the bisector of the angle PAQ be (α,β)(\alpha,\beta). Then, the value of 7α+3β7\alpha+ 3\beta is equal to

  3. Question 3

    8 Apr 2024

    177
    Let a ray of light passing through the point (3,10)(3,10) reflects on the line 2x+y=62x+y= 6 and the reflected ray passes through the point (7,2)(7,2). If the equation of the incident ray is ax+by+1=0ax + by + 1 = 0, then a2+b2+3aba^{2}+b^{2}+ 3ab is equal to

  4. Question 4

    9 Apr 2024

    64
    A ray of light coming from the point P(1,2)P(1,2) gets reflected from the point QQ on the xx-axis and then passes through the point R(4,3)R(4,3). If the point SS (h, kk ) is such that PQRSPQRS is a parallelogram, then hk2hk^{2} is equal to:

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