# Vector equation and curvature

Vector equation

$r(t)=2\cos(t)\mathbf i + 3\sin(t)\mathbf j\ \ (0 \le t \le 2\pi)$

represents ellipse.

I need to find curvature of this ellipse on endpoints of x and y axis that are given with $(2,0),(-2,0)$ and $(0,3),(0,-3)$ respectively.

Work so far:

• I need to evaluate curvature $k$ at $t=0$ and $t=\pi/2$
• $k = {{x'y''-y'x''}\over{(x'^2 + y'^2)^{3/2}}}$
• x and y for plugging to above equation are $x(t)=2\cos(t)$ and $y=3\sin(t)$
• After differentiating we have $k = {{6}\over{(5\cos(t)+4)^{3/2}}}$
• After evaluating for $t=0$ and $t=\pi/2$ we have $2\over9$ and $3\over4$ respectively
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Use formula 13 here. –  Ｊ. Ｍ. Oct 22 '11 at 8:29
That ain't much work; you still remember how to differentiate a sine and a cosine, no? –  Ｊ. Ｍ. Oct 22 '11 at 9:24
Are you sure about that formula you got for $k$? You seem to have lost a square in there... –  Ｊ. Ｍ. Oct 22 '11 at 12:25