Question

Solve:

$a{(x-2y+1)}^2+9{(2x+y+2)}^2=25$

Answer 1

We have ${(a+b+c)}^2=a^2+b^2+c^2+2ab+2bc+2ca$

Using this, we have

$a(x^2+4y^2+1-4xy-4y+2x)+9(4x^2+y^2+4+4xy+4y+8x)-25=0$

$\Rightarrow (a{x}^2+36x^2)+(4a{y}^2+9y^2)+(a+36)+(-4axy+36xy)+(-4ay+36y)+(2ax+72x)-25=0$

$\Rightarrow (a+36)x^2+(4a+9)y^2+(a+36-25)+(-4a+36)xy+(-4a+36)y+(2a+72)x=0$

$\Rightarrow (a+36)x^2+(4a+9)y^2+(a+11)-4(a-9)xy-4(a-9)y+2(a+36)x=0$

$\Rightarrow (a+36)x^2+(4a+9)y^2-4(a-9)(xy+x)+2(a+36)x+(a+11)=0$

$\Rightarrow (a+36)x^2+(4a+9)y^2-4x(a-9)(y+1)+2(a+36)x+(a+11)=0$