Determine the maximum area of the circle which is entirely contained within the parabola y^{2} = 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 20071231 12:26:12 