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Xaphene
:
Find eccentricity and length of latus rectum of the ellipse x^(2)+2y^(2)=3.
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May 27, 2014 at 1:49pm
Little Sky
:
Ellipse general formula:[deqn]\left( \frac{x}{a} \right)^2+\left( \frac{y}{b} \right)^2=1[/deqn]Eccentricity:[deqn]e=\sqrt{1-\left( \frac{b}{a} \right)^2}[/deqn]Length of latus rectum:[deqn]l=\frac{2b^2}{a}[/deqn]By modifying the ellipse equation,[deqn]\frac{x^2}{3}+\frac{2y^2}{3}=\frac{3}{3}\\\frac{1}{3}x^2+\frac{2}{3}y^2=1\\\frac{1}{a^2}=\frac{1}{3};\frac{1}{b^2}=\frac{2}{3}\\a=\sqrt{3};b=\sqrt{\frac{3}{2}}[/deqn]Eccentricity:[deqn]e=\sqrt{1-\left( \frac{\sqrt{\frac{3}{2}}}{\sqrt{3}} \right)^2}\\=\sqrt{1-\frac{\frac{3}{2}}{3}}~~~~~~~~~~\\=\sqrt{\frac{1}{2}}~~~~~~~~~~~~~~~~\\=\frac{1}{\sqrt{2}}~~~~~~~~~~~~~~~~~[/deqn]Length of latus rectum:[deqn]l=\frac{2\left( \sqrt{\frac{3}{2}} \right)^2}{\sqrt{3}}\\=\frac{2\left( \frac{3}{2} \right)}{\sqrt{3}}~~~~~~~~\\=\frac{3}{\sqrt{3}}~~~~~~~~~~~~\\=\sqrt{3}~~~~~~~~~~~~~~~[/deqn]
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May 28, 2014 at 11:29pm
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