Question

Which of the following is a tangent to the curve given by $x^3+y^3=2xy?$

• A. $y=x$
• B. $y=x+2$
• C. $y=-x+2$
• D. $y=-x+3$

Answer 1

Answer: (A)

Given curves is $x^3+y^3=2xy$ ..(i)
If a line is a tangent of given curve, then it will touch at only one point.
For $y=x$ to be a tangent,
$x^3+x^3=2x\times x\Rightarrow 2x^3=2x^2$
$\Rightarrow$ $x=1$ On putting the value of x in given curve (i), we get ${(1)}^3+y^3=2\times 1\times y$
$\Rightarrow$ $1+y^3=2y$
$\Rightarrow$ $y^3-2y+1=0$
$\Rightarrow$ $y^3-y^2+y^2-2y+1=0$
$\Rightarrow$ $y^2(y-1)+y^2-y+y-2y+1=0$
$\Rightarrow$ $y^2(y-1)+y(y-1)-y+1=0$
$\Rightarrow$ $y^2(y-1)+y(y-1)-1(y-1)=0$
$\Rightarrow$ $(y-1)(y^2+y-1)=0$
$\Rightarrow$ $y-1=0$ or $y^2+y-1=0$
$\Rightarrow$ $y=1$ or $y=\frac{-1\pm \sqrt{1-4\times 1\times (-1)}}{2}$
$\Rightarrow$ $y=1$ or $y=\frac{-1\pm \sqrt{5}}{2}$ [not integral value] So, $(x,y)=(1,1)$
Then, $y=x$ will be the tangent to the given curve.