Question

If $A$ represents the area of the ellipse $3x^2+4xy+3y^2=1$, then the value of $\frac{3\sqrt{5}}{\pi }A$ is _____.

Answer 1

The Area of are Ellipse is $\pi$ as

If we replace y by x, into the equation of

Ellipse we get same,

$y\to x$ & $x\to y.$

$3x^2+3y^2+4xy=1\Rightarrow$ Symmetric about $y=x$

If we replace in following

$y\to -x$ & $x\to -y$

$3x^2+4xy+3y^2=1\Rightarrow$ Symmetric about $y=-x$

from above

Ellipse is centered at origin, with

it's axis as $y=x$ & $y=-x$

Now, put $y_1=x$ into ellipse equation

$3x^2+4x^2+3x^2=1$

$x=\pm \frac{1}{\sqrt{10}}$

point of intersection $(\frac{1}{\sqrt{10}},\frac{1}{\sqrt{10}})$ & $(-\frac{1}{\sqrt{10}},-\frac{1}{\sqrt{10}})$

Now, put $y=-x$ into ellipse eq.

$3x^2-4x^2+3x^2=1$

$x=\pm \frac{1}{\sqrt{2}}$

point of intersection $(\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}})$ & $(-\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}})$

So Distance between are of pair is a, & Distance b/w

other pair is b. So $2a=\frac{2}{\sqrt{5}}$ & $2b=2$

$A=\pi ab.=\pi \times \frac{1}{\sqrt{5}}\times \frac{1}{1}=\frac{\pi }{\sqrt{5}}$

given $\frac{3\sqrt{5}}{\pi }\times A=\frac{3\sqrt{5}}{\pi }\times \frac{\pi }{\sqrt{5}}=3$