Question

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

• A. $\left(0,1\right)$
• B. $\left(1,1\right)$
• C. $\left(0,0\right)$
• D. $\left(1,0\right)$

Answer 1

The correct option is D

$\left(1,0\right)$

Explanation for correct option:

Finding the points:

Given that the equation of the given curve is

$x+y=e^{xy}$

The tangents is parallel to the $y$- axis .

So, the slope of the tangents will be infinity.

$\therefore \frac{dy}{dx}=\infty$

Calculating $\frac{dy}{dx}$:

$x+y=e^{xy}$

Differentiating both sides w.r.t '$x$'.

$\begin{array}{c}1+\frac{dy}{dx}=e^{xy}\left(y+x.\frac{dy}{dx}\right)\\ \frac{dy}{dx}-x.e^{xy}\frac{dy}{dx}=e^{xy}y-1\\ \frac{dy}{dx}\left(1-x{e}^{xy}\right)=e^{xy}y-1\\ \frac{dy}{dx}=\frac{e^{xy}y-1}{1-x{e}^{xy}}\end{array}$

Now $\frac{dy}{dx}=0$

$\therefore 1-x{e}^{xy}=0$

$\Rightarrow 1=x{e}^{xy}$

Checking the truth of $1-x{e}^{xy}=0$ for different options..

Option (A) $\left(0,1\right)$

$$\begin{gathered} L.H.S=0 \\ R.H.S=1-0e^{0\times 1} \\ =1 \\ L.H.S\ne R.H.S \end{gathered}$$

Therefore option (A) is not the correct answer.

Option (B) $\left(1,1\right)$

$$\begin{gathered} L.H.S=0 \\ R.H.S=1-1e^{1\times 1} \\ =1-e \\ L.H.S\ne R.H.S \end{gathered}$$

Therefore option (B) is not the correct answer.

Option (C) $\left(0,0\right)$

$$\begin{gathered} L.H.S=0 \\ R.H.S=1-0e^{0\times 0} \\ =1 \\ L.H.S\ne R.H.S \end{gathered}$$

Therefore option (C) is not the correct answer.

Option (D) $\left(1,0\right)$

$\begin{array}{rcl}LHS& =& 0\\ RHS& =& 1-1e^{1\times 0}\\ & =& 1-1\\ & =& 0\\ LHS& =& RHS\end{array}$

Hence, option (D) is correct.