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Expressions in the sum and product of the slopes of a homogeneous pair

This type of question, asked 4 times: 9 May 2024 S2, 19 Apr 2026 S2, 20 Apr 2026 S2, 14 May 2026 S2. All 4 questions below, oldest first.

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  1. Question 1

    9 May 2024 S2

    101
    If the slope of one of the lines represented by ax2+(2a+1)xy+2y2=0ax^2 + (2a+1)xy + 2y^2 = 0 is reciprocal of the slope of the other, then the sum of squares of slopes is

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  2. Question 2

    19 Apr 2026 S2

    55
    If two lines represented by x2−(1+3)xy+3y2=0x^2 - (1 + \sqrt{3})xy + \sqrt{3}y^2 = 0 make angles α\alpha and β\beta with the X-axis, then tan⁡(α+β)\tan(\alpha + \beta) is...

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  3. Question 3

    20 Apr 2026 S2

    56
    If the angle made by the lines represented by the equation ax2+2hxy+by2=0ax^2 + 2hxy + by^2 = 0 with X-axis are α\alpha and β\beta, then tan⁡(α+β)\tan(\alpha + \beta) is

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  4. Question 4

    14 May 2026 S2

    60
    If the sum of the cubes of slopes of two lines represented by the equation ax2+2hxy+by2=0ax^2 + 2hxy + by^2 = 0 is 11, then

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