The figure above shows two ellipses whose major axes are perpendicular to each other. Each ellipse passes through the other ellipse’s foci, which form the vertices of a square. If the shaded square encloses an area of 16, then what is the area enclosed by one of the ellipses?
Find the shortest distance between the parabolas and .
Note: Round off your answer to 2 decimal places.
are consecutive vertices of a rectangle whose area is square units. An ellipse with area , which passes through and has its foci at and .
If the perimeter of the rectangle can be expressed as where is a positive integer and is a square-free positive integer, find .
Consider an ellipse . A line is drawn tangent to the ellipse at a point . A line segment drawn from the origin to a point on this line is perpendicular to this tangent line.
Find the maximum area of .
The area of a circle centered at the origin, which is inscribed in the parabola can be expressed as where and are coprime positive integers. What is the value of ?