If the sum of the distances from a variable point P to the given points A ( 1 , 0 ) and B ( 0 , 1 ) is 2 ,…

It is given that, $AP+BP=2$

Consider the point $P$ be$(x, y)$.

$\sqrt{(x-1{)}^2+y^2}+\sqrt{x^2+(y-1{)}^2}=2$

$\Rightarrow \sqrt{(x-1{)}^2+y^2}=2-\sqrt{x^2+(y-1{)}^2}$

Squaring on both sides,

$(x-1{)}^2+y^2=4+x^2+(y-1{)}^2-4\sqrt{x^2+(y-1{)}^2}$

$\Rightarrow x^2+y^2+1-2x=4+x^2+y^2+1-2y-4\sqrt{x^2+(y-1{)}^2}$

$\Rightarrow 2y-2x=4-4\sqrt{x^2+(y-1{)}^2}$

$\Rightarrow y-x-2=-2\sqrt{x^2+(y-1{)}^2}$

Squaring both the sides,

$x^2+y^2+4-2xy+4x-4y=4x^2+4y^2+4-8y$

$\Rightarrow 3x^2+3y^2-4x-4y+2xy=0$