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

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
$\hspace{-0.5cm}Given equation of curve is 3x^2+4xy+4y^2=1\;, Now put x=u+v and \\\\y=u-v. So we get 10u^2+2v^2=1\Rightarrow \frac{x^2}{\left(\frac{1}{\sqrt{10}}\right)^2}+\frac{y^2}{\left(\frac{1}{\sqrt{2}}\right)^2} = 1\\\\Now we know that bounded area is Independent of Shifting.\\\\So area bounded by 3x^2+4xy+3y^2=1 is same as area enclosed by New \\\\Ellipse.$

$\hspace{-0.5cm}So Area of Ellipse whose equation is \frac{x^2}{a^2}+\frac{y^2}{b^2} = 1. is \pi ab\; sqrt. unit.\\\\ So Here a=\frac{1}{\sqrt{10}}\;\;,b=\frac{1}{\sqrt{2}}. So Bounded area is A=\pi\cdot \frac{1}{\sqrt{10}}\cdot \frac{1}{\sqrt{2}} = \frac{\pi}{2\sqrt{5}}\\\\ So we get A=\frac{3\sqrt{5}A}{\pi} = \frac{3\sqrt{5}}{\pi}\times \frac{\pi}{2\sqrt{3}} = \frac{3}{2}.$