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Slopes of the lines of a given pair are in a given ratio: find the unknown coefficient

The same question, asked 4 times: 2 May 2023 S1*, 11 May 2023 S1*, 15 May 2023 S1*, 25 Apr 2025 S2. Every wording below, oldest first.

Try each one first, then open its answer.

  1. Question 1

    2 May 2023 S1

    107
    The slopes of the lines given by x2+2hxy+2y2=0x^2 + 2hxy + 2y^2 = 0 are in the ratio 1:21:2, then hh is

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  2. Question 2

    11 May 2023 S1

    142
    If the slope of one of the lines given by Kx2+6xy+y2=0Kx^2 + 6xy + y^2 = 0 is three times the other, then the value of K is

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  3. Question 3

    15 May 2023 S1

    147
    The value of kk, if the slope of one of the lines given by 4x2+kxy+y2=04x^{2}+kxy+y^{2}=0 is four times that of the other, is given by

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  4. Question 4

    25 Apr 2025 S2

    108
    The slopes of the lines represented by 6x2+2hxy+y2=06x^{2}+ 2hxy+y^{2}= 0 are in the ratio 2:32:3, then h=h =

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