Question

Find the separate equation of the lines represented by the following equation:

$(x-2{)}^2-3(x-2)(y+1)+2(y+1{)}^2=0$

Answer 1

${(x-2)}^2-3(x-2)(y+1)+2{(y+1)}^2=0\Rightarrow {(x-2)}^2-2\times \frac{3}{2}(x-2)(y+1)+\frac{9}{4}{(y+1)}^2-\frac{1}{4}{(y+1)}^2=0\Rightarrow {(x-2-\frac{3}{2}(y+1\left)\right)}^2-\frac{1}{4}{(y+1)}^2=0\Rightarrow {\frac{(2x-4-3(y+1\left)\right)}{4}}^2-\frac{1}{4}{(y+1)}^2=0\Rightarrow (2x-4-3y-3-y-1)(2x-4-3y-3+y+1)=0\Rightarrow 2x-4y-8=0or2x-2y-6=0\Rightarrow x-2y-4=0orx-y-3=0$

The separate equations of the lines are $x-2y=4$ and $x-y=3$