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A line dividing a region's area: find the ratio or the line

This type of question, asked 3 times: 2021 Paper 23, 14 Jun 2022, 22 Jan 2026 S1. All 3 questions below, oldest first.

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  1. Question 1

    2021 Paper 23

    88
    If the line y=mxy=mx bisects the area enclosed by the lines x=0,y=0,x=32x = 0,y = 0,x =\frac{3}{2} and the curve y=1+4x−x2y= 1 + 4x-x^{2}, then 12 m12\text{ }m is equal to

  2. Question 2

    14 Jun 2022

    89
    Let SS be the region bounded by the curves y=x3y =x^{3} and y2=xy^{2}= x. The curve y=2∣x∣y = 2|x| divides SS into two regions of areas R1R_{1} and R2R_{2}. If max⁡{R1,R2}=R2\max\left\{ R_{1},R_{2} \right\}=R_{2}, then R2R1\frac{R_{2}}{R_{1}} is equal to

  3. Question 3

    22 Jan 2026 S1

    53
    Let the line x=−1x = - 1 divide the area of the region {(x,y):1+x2≤y≤3−x}\left\{ (x,y):1 +x^{2}\leq y \leq 3 - x \right\} in the ratio m:n,gcd(m,n)=1m:n,gcd(m,n) = 1. Then m+nm + n is equal to

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