Question

The curve given by $x+y=e^{xy}$ has a tangent as the $y$-axis at the point

• A. $(0,1)$
• B. $(1,0)$
• C. $(1,1)$
• D. none of these

Answer 1

Answer: (B)

$(1,0)$

We have, $x+y=e^{xy}\Rightarrow \left(1\right)$
Differentiate both sides w.r.t. $x$ we get,
$1+\frac{dy}{dx}=e^{xy}(y+x\frac{dy}{dx})$
$\Rightarrow \frac{dy}{dx}=\frac{y{e}^{xy}-1}{1-x{e}^{xy}}$
Now for tangent to be y-axis, $\left|\frac{dy}{dx}\right|\to \infty$
$\Rightarrow 1-x{e}^{xy}=0\Rightarrow \left(2\right)$
Solving (1) and (2) we get, $(x,y)=(1,0)$, which is the required point