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Maximum area of a rectangle or triangle inscribed under a parabola

This type of question, asked 3 times: 30 Jan 2024, 8 Apr 2024, 24 Jan 2025. All 3 questions below, oldest first.

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  1. Question 1

    30 Jan 2024

    65
    The maximum area of a triangle whose one vertex is at (0,0)(0,0) and the other two vertices lie on the curve y=−2x2+54y = - 2x^{2}+ 54 at points (x,y(x,y ) and (−x,y)( - x,y) where y>0y> 0 is:

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  2. Question 2

    8 Apr 2024

    171
    Let AA be the region enclosed by the parabola y2=2xy^{2}= 2x and the line x=24x= 24. Then the maximum area of the rectangle inscribed in the region AA is.

  3. Question 3

    24 Jan 2025

    62
    Consider the region R={(x,y):x≤y≤9−113x2,x≥0}R =\left\{ (x,y):x \leq y \leq 9 -\frac{11}{3}x^{2},x \geq 0 \right\}. The area, of the largest rectangle of sides parallel to the coordinate axes and inscribed in RR, is :

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