Find the area of region enclosed by curve 
5x​2 + 6xy + 2y2 + 7x + 6y + 6 =0 .

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Please find below the solution to the asked query:

This question requires great calculations.As pointed by other student, equation comes out to be ellipse.We have5x2+6xy+2y2+7x+6y+6=0....iax2+bxy+cy2+dx+ey+f=0We will apply rotation of conics here.tan2θ=ba-corcotθ=a-cb+1+a-cb2=5-26+1+5-262=12+52=1+52cosecθ=1+cot2θ=1+1+522=4+1+5+252=10+252sinθ=210+25cosθ=1-sin2θ=1-410+25=10+25-410+25=6+2510+25x=Xcosθ-Ysinθy=Xsinθ+Ycosθ, where cosθ,sinθ are already knownOnce you replace value of x and y in i, xy term will cancel out andafter making perfect squares, you will getX-α2A2+Y-β2B2=1Area of ellipse=πABAnswer will be π2

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