Question

If $x^3+y^3+1=3xy$ represents a vertex and a side of a triangle then locus of the centroid of the triangle will be.

• A. $x-y=1$
• B. $x-y=0$
• C. $x+y=1$
• D. $x+y=0$

Answer 1

The correct option is D $x+y=0$

Given that $x^3+y^3+1=3xy$ represent a vertex and a side of triangle

$x^3+y^3+1=3xy-1$

taking denvative cost $x$

$3x^2+3y^2\frac{dy}{dx}=3[x\frac{dy}{dx}+y]\Rightarrow 3x^2+3y^2\frac{dy}{dx}=3x\frac{dy}{dx}+3y(6y^2-3x)\frac{dy}{dx}=3y-3x^2(y^2-x)\frac{dy}{dx}=y-x^2\Rightarrow \frac{dy}{dx}=\frac{y-x^2}{y^2-x}\Rightarrow \frac{dy}{dx}=0\Rightarrow y^2-x^2=0(y-x)(y+x)=0x+y=0$