Area of a pillow
The graphs of four identical parabolas, as shown above, intersect at the points
and
. The vertices of these parabolas are all
units away from the origin, and the area enclosed by the quadrilateral whose vertices are the foci of these parabolas is
square units.
If the area formed about the origin enclosed by the four parabolas can be expressed in the form , where and are coprime positive integers, determine .
Details and Assumptions:
- The parabolas are identical in the sense that they have the same focal lengths.
The answer is 223.
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2 solutions
Moderator note:
That is indeed a brute force approach. Given that you have found the equation of the parabola, is would be slightly more straightforward to find the area under the curve.
A sort of brute force option is to parameterize a couple of the parabolas and use Green's theorem to determine the area. Since 3 points determine a parabola opening up or opening to the side, and since the foci are on the x- and y-axes, the parametrization is straightforward. For example, parabola1[t ] := {0, 4} + {t, t^2/25}, and parabola2[t ] := {4, 0} + {t^2/25, t}.
Integrating the curves along the closed spaced defined by these curves and the y- and x-axis in a counterclockwise direction in an appropriate vector field gives the area of this pillow quadrant, and multiplying by 4 gives the result.