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If A represents the area of the ellipse 3x^2+4xy +3y^2=1, then the value of 3√5A/pi is

If A represents the area of the ellipse 3x^2+4xy +3y^2=1, then the value of 3√5A/pi is

Grade:12th pass

2 Answers

Yash Arora
41 Points
8 years ago
area of ellipse= pie * a*b
 
a = (1/3)^2 = b
 
A = 3√5×1/3×pie/pie
 
 
A= root 5
 
 
 
 
do check for the formula yourself
jagdish singh singh
173 Points
8 years ago
\hspace{-0.5cm}$Given equation of curve is $3x^2+4xy+4y^2=1\;,$ Now put $x=u+v$ and \\\\$y=u-v.$ So we get $10u^2+2v^2=1\Rightarrow \frac{x^2}{\left(\frac{1}{\sqrt{10}}\right)^2}+\frac{y^2}{\left(\frac{1}{\sqrt{2}}\right)^2} = 1$\\\\Now we know that bounded area is Independent of Shifting.\\\\So area bounded by $3x^2+4xy+3y^2=1$ is same as area enclosed by New \\\\Ellipse.
 
\hspace{-0.5cm}$So Area of Ellipse whose equation is $\frac{x^2}{a^2}+\frac{y^2}{b^2} = 1.$ is $\pi ab\; $ sqrt. unit.\\\\ So Here $a=\frac{1}{\sqrt{10}}\;\;,b=\frac{1}{\sqrt{2}}.$ So Bounded area is $A=\pi\cdot \frac{1}{\sqrt{10}}\cdot \frac{1}{\sqrt{2}} = \frac{\pi}{2\sqrt{5}}$\\\\ So we get $A=\frac{3\sqrt{5}A}{\pi} = \frac{3\sqrt{5}}{\pi}\times \frac{\pi}{2\sqrt{3}} = \frac{3}{2}.$

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