Determine the maximum area of the circle which is entirely contained within the parabola y² = 36x, and passes through its focus.

The parabola,

   y^2 = 36x = 4px,

where p is the distance from the vertex to the focus.

The circle,

   (x - h)^2 + y^2 = r^2.

Since the circle passes throught the focus,

   (p - h)^2 + 0^2 = r^2.

Therefore,

   (x - h)^2 + y^2 = (p - h)^2.

The intersection of the circle and ellipse,

   (x - h)^2 + 4px = (p - h)^2

              or

   x^2 - 2(h - 2p)x - (p^2 - 2ph) = 0.

Solving for x,
        
   x = h - 2p +- sqrt[(h - p)(h - 5p)].

If the radicand is zero, then the circle is
tangent to the ellipse and we have the largest
circle within the ellipse.

h = p implies the radius of the circle is zero.

Therefore, h = 5p and r^2 = (4p)^2.

For our problem,

   Area of circle = PI*(36)^2.

Posted by Bractals on 2007-12-31 12:26:12