Question

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

• A. \( (0,1) \)
• B. \( (1,0) \)
• C. \( (1,1) \)
• D. None of these

Hint:

For finding points where the tangent is parallel to the Y-axis: - Compute \( \frac{dy}{dx} \). - Identify where the denominator of \( \frac{dy}{dx} \) is zero. - Check the given points to find a valid solution.

Answer 1

The Correct Option is B

Step 1: Differentiate the given equation implicitly. Given: \[ x + y = e^{xy} \] Differentiating both sides with respect to \( x \), using implicit differentiation: \[ \frac{d}{dx} (x + y) = \frac{d}{dx} (e^{xy}) \] Applying differentiation: \[ 1 + \frac{dy}{dx} = e^{xy} \cdot \left( x \frac{dy}{dx} + y \right) \] Rearrange to express \( \frac{dy}{dx} \): \[ 1 + \frac{dy}{dx} = e^{xy} (x \frac{dy}{dx} + y) \] \[ 1 + \frac{dy}{dx} - e^{xy} y = e^{xy} x \frac{dy}{dx} \] \[ 1 - e^{xy} y = \frac{dy}{dx} (e^{xy} x - 1) \] \[ \frac{dy}{dx} = \frac{1 - e^{xy} y}{e^{xy} x - 1} \] Step 2: Condition for a tangent parallel to the Y-axis. A tangent is parallel to the Y-axis when \( \frac{dx}{dy} = 0 \), which means \( \frac{dy}{dx} \) is undefined. For \( \frac{dy}{dx} \) to be undefined, the denominator must be zero: \[ e^{xy} x - 1 = 0 \] \[ e^{xy} x = 1 \] \[ x = e^{-xy} \] Step 3: Check given Option. For \( (1,0) \): \[ x = 1, \quad y = 0 \] \[ e^{(1)(0)} \cdot 1 = 1 \] \[ 1 = 1 \quad {(satisfied)} \] Thus, the correct point is \( (1,0) \).