Question

Locus of all point $P\left(x,y\right)$ satisfy $x^3+y^3+3xy=1$ consist a 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

Explanation for the correct options:

Evaluate the union of the given locus

Given, $x^3+y^3+3xy=1$

$\Rightarrow x^3+y^3+{\left(-1\right)}^3-3xy\left(-1\right)=0$

Using the identity $a^3+b^3+c^3-3abc=\left(a+b+c\right)\left(a^2+b^2+c^2-ab-ac-bc\right)$

$$\begin{gathered} \Rightarrow \left(x+y-1\right)\left(x^2+y^2+1-xy+y+x\right)=0 \\ \Rightarrow \left(x+y-1\right)\left(2x^2+2y^2+2-2xy+2y+2x\right)=0\mathbf{[}\mathbf{multiply}\mathbf{}\mathbf{by}\mathbf{}\mathbf{2}\mathbf{]} \\ \Rightarrow \left(x+y-1\right)\left({\left(x-y\right)}^2+{\left(x+1\right)}^2+{\left(y+1\right)}^2\right)=0 \end{gathered}$$

$\Rightarrow x+y=1$ or $\left({\left(x-y\right)}^2+{\left(x+1\right)}^2+{\left(y+1\right)}^2\right)=0\Rightarrow x=y=-1$

So, there's a line $x+y=1$ or point $\left(-1,-1\right)$

Thus, the given locus denotes a line and a isolated point.

Hence, the correct option is (A)