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)

$y=x$

Answer 2

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,
${{x}^{3}}+{{x}^{3}}=2x\times x\Rightarrow 2{{x}^{3}}=2{{x}^{2}}$
$\Rightarrow$ $x=1$ 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.