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Find a parameter from a given relation between two areas

This type of question, asked 3 times: 27 Jun 2022, 25 Jan 2023, 28 Jan 2026 S2. All 3 questions below, oldest first.

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  1. Question 1

    27 Jun 2022

    86
    Let A1={(x,y):∣x∣≤y2,∣x∣+2y≤8}A_{1}=\left\{ (x,y):|x| \leq y^{2},|x| + 2y\leq 8 \right\} and A2={(x,y):∣x∣+∣y∣≤k}A_{2}= \{(x,y):|x| + |y| \leq k\}. If 27(27\left( \right. Area  A1)=5\left. \ A_{1} \right)= 5 (Area A2A_{2} ), then kk is equal to:

  2. Question 2

    25 Jan 2023

    89
    If the area enclosed by the parabolas P1:2y=5x2P_{1}:2y = 5x^{2} and P2:x2−y+6=0P_{2}:x^{2}- y + 6 = 0 is equal to the area enclosed by P1P_{1} and y=αx,α>0y = \alpha x,\alpha > 0, then α3\alpha^{3} is equal to

  3. Question 3

    28 Jan 2026 S2

    61
    Let P1:y=4x2P_{1}:y= 4x^{2} and P2:y=x2+27P_{2}:y=x^{2}+ 27 be two parabolas. If the area of the bounded region enclosed between P1P_{1} and P2P_{2} is six times the area of the bounded region enclosed between the line y=αx,α>0y =\alpha x,\alpha> 0 and P1P_{1}, then α\alpha is equal to :

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