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A line touches y^2 = px^3 + q at a given point: find p and q

The same question, asked 4 times: 10 May 2023 S2*, 14 May 2024 S2*, 17 Apr 2026 S2*, 20 Apr 2026 S2. Every wording below, oldest first.

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  1. Question 1

    10 May 2023 S2

    119
    If y=4x−5y = 4x-5 is a tangent to the curve y2=px3+qy^2 = px^3+q at (2,3)(2,3), then the values of p and q are respectively

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  2. Question 2

    14 May 2024 S2

    130
    If y=4x−5y = 4x - 5 is a tangent to the curve y2=px3+qy^2 = px^3 + q at (2,3)(2,3), then p−qp - q is

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  3. Question 3

    17 Apr 2026 S2

    76
    If the line y=4x−5y = 4x - 5 is tangent to the curve y2=ax3+by^2 = ax^3 + b at the point (2,3)(2, 3), then the value of 7a−2b7a - 2b is...

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  4. Question 4

    20 Apr 2026 S2

    76
    The line x+y=0x + y = 0 touches the curve y2=ax3+by^2 = ax^3 + b at (1,−1)(1, -1) then values of aa and bb respectively are ...........

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