# A geometry student wants to draw a rectangle inscribed in the ellipse x2+4y2=49. What is the area of the largest reactangle that the student can draw?

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A geometry student wants to draw a rectangle inscribed in the ellipse x2+4y2=49. What is the area of the largest reactangle that the student can draw?

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Step 1

For maximization, equate the derivative of the function to 0 to get the critical points.

Then check the value of function at these critical points. If function is negative then the critical point is the point of maxima. If it is positive, then it is minima otherwise the test fails.

Step 2

For largest area all of the points of the rectangle will touch the ellipse.

Assume any point (x,y) on the ellipse.

Then other horizontal end of one side will be (-x,y) and vertical end of one side will be (x,-y).

So, length of rectangle =2x

Thus, A=(2x)(2y)

=4xy

Step 3

Derive the value of y from th...

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