Question

Locus of all point $P(x,y)$ satisfying $x^3+y^3+3xy=1$ consists of union of

• A. a line and an isolated point
• B. a line pair and an isolated point
• C. a line and a circle
• D. a circle and an isolated point

Answer 1

The correct option is A a line and an isolated point

$x^3+y^3+3xy=1$
$x^3+y^3+(-1{)}^3-3xy(-1)=0$
using the identity $a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-bc-ca)$
$\Rightarrow (x+y-1)(x^2+y^2+1-xy+y+x)=0$
$\to (x+y-1)(2x^2+2y^2+2-2xy+2y+2x)=0$
$\to (x+y-1)\left[\right(x-y{)}^2+(x+1{)}^2+(y+1{)}^2]=0$
$\Rightarrow x+y=1$ or $x=y=-1$
$\Rightarrow$ a line $x+y-1=0$ or a point $(-1,-1)$
$\therefore$ the given locus represents a line and an isolated point.
Hence, option $A$ is correct.