Rectangle inscribed on two parabolas but not parallel to the axisLet f ( x ) = 2 x 2 and g ( x ) = 36 − x 2 . Let R be the closed region between them. Draw a rectangle ABCD such that A and B are points on f(x), and C and D are points on g(x), and the rectangle is not parallel to the axis, or prove that to be impossible.

Question

Rectangle inscribed on two parabolas but not parallel to the axisLet   f  (  x  )  =  2      x    2   and   g  (  x  )  =  36  −      x    2  . Let R be the closed region between them. Draw a rectangle ABCD such that A and B are points on f(x), and C and D are points on g(x), and the rectangle is not parallel to the axis, or prove that to be impossible.

betoosolis7i · Accepted Answer

Step 1Brute force:First, we relax the condition to finding any parallelogram with 2 vertices on each equation.Let   A  =  (  a  ,  2      a    2    )  ,  B  =  (  a  +  b  ,  2      a    2    +  4  a  b  +  2      b    2    )  ,  D  =  (  d  ,  36  −      d    2    )which makes   C  =  (  d  +  b  ,  36  −  (  d  +  b      )    2    =  36  −      d    2    +  4  a  b  +  2      b    2    ). This gives   b  =  0 (rejected),   d  =  −  2  a  −            3      b        2  .Now, we to make the parallelogram into a rectangle, we require   A  B  ⊥  A  D  ⇒  (  b  ,  4  a  b  +  2      b    2    )  ⋅  (  d  −  a  ,  36  −      d    2    −  2      a    2    )  =  0. This simplifies to   −  3      /    2  b  (  2  a  +  b  )  (  8      a    2    +  8  a  b  +  3      b    2    −  47  )  =  0.Step 2Note that   b  =  0 is rejected, and   2  a  +  b means AB is parallel to the x-axis, hence rejected. So, we&#039;re looking for solutions to   8      a    2    +  8  a  b  +  3      b    2    −  47  =  0, which is an ellipse.As it turns out,   a  =  −  3  ,  b  =  4  −            23      3       is one such solution, which leads to the rectangle.  A  =  (  −  3  ,  18  )  ,  B  ≈  (  −  1.77  ,  6.26  )  ,  C  ≈  (  5.38  ,  7.01  )  ,  D  ≈  (  4.15  ,  18.75  )Of course, if you don&#039;t want A on the intersection point, take   a  =  −  3  +  ϵ

Let f(x)=2x^2 and g(x)=36-x^2. Let R be the closed region between them. Draw a rectangle ABCD such that A and B are points on f(x), and C and D are points on g(x), and the rectangle is not parallel to the axis, or prove that to be impossible.

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