If A represents the area of the ellipse $3 x^{2} + 4 x y + 3 y^{2} = 1$ , then the value of $\frac{3 \sqrt{5}}{π}$ A is __________?

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Solution

If an ellipse is defined implicity by
$α x^{2} + β x y + γ y^{2} = 1$
Then its area $A = \frac{2 π}{\sqrt{4 α γ - β^{2}}}$
The equation is $3 x^{2} + 4 x y + 3 y^{2} = 1$
Here $A = \frac{2 π}{\sqrt{4.3.3 - 16}}$
$A = \frac{2 π}{\sqrt{20}}$
$\frac{3 \sqrt{5}}{π} A = \frac{3 \sqrt{5}}{π} \times \frac{2 π}{\sqrt{20}}$
$= \frac{3 \sqrt{5}}{π} \times \frac{2 π}{2 \sqrt{5}}$
$= 3$

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