area of largest rectangle in ellipse, closest pt on parabola

[des4ij]

1) Find the area of the largest rectangle that can be inscribed in the ellipse below.

x^2/a^2 + y^2/b^2 = 1

2) Find the point on the parabola that is closest to the point (0, -3).

x + y^2 = 0

Thx... any help appreciated...

 

[galactus]

Re: largest area in an ellipse...

Find the area of the largest rectangle that can be inscribed in the ellipse below.

x^2/a^2+y^2/b^2=1

Check this link:

http://www.freemathhelp.com/forum/viewt ... ht=ellipse

ALso,

Find the point on the parabola that is closest to the point (0,-3).

x + y^2 = 0

Thx... any help appreciated...

\(\displaystyle \L\\y=\pm\sqrt{-x}\)

\(\displaystyle \L\\L=\sqrt{(x-0)^{2}+(y+3)^{2}}\)

\(\displaystyle \L\\L=\sqrt{x^{2}+(-\sqrt{-x}+3)^{2}}\)

The distance and the square of the distance both have their max and mins at the same place, thus, eliminating the need for a radical.

\(\displaystyle \L\\S=L^{2}=x^{2}-6\sqrt{-x}-x+9\)

\(\displaystyle \L\\\frac{dS}{dx}=\frac{3}{\sqrt{-x}}+2x-1\)

Set \(\displaystyle \frac{dS}{dx}=0\), solve for x. Use that value to find your corresponding y value and, therefore, the needed point.

 

[des4ij]

thx got it...