Problem 18 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The curve \(y-e^{x y}+x=0\) has a vertical tangent at the point (A) \((1,1)\) (B) at no point (C) \((0,1)\) (D) \((1,0)\)

Step-by-Step Solution

Answer

The curve has a vertical tangent at the point (0, 1).

Step 1: Understand the Problem

We need to determine the point on the curve \(y - e^{xy} + x = 0\) where the tangent is vertical. A vertical tangent means that the derivative \(\frac{dy}{dx}\) is undefined (infinite slope).

Step 2: Implicitly Differentiate the Equation

Implicitly differentiate the equation \(y - e^{xy} + x = 0\) with respect to \(x\). We get: \[\frac{dy}{dx} - \left(e^{xy} \cdot (y + x \cdot \frac{dy}{dx})\right) + 1 = 0\]. Simplify this to find \(\frac{dy}{dx}\).

Step 3: Simplify and Solve for dy/dx

Rearrange the result of differentiation to solve for \(\frac{dy}{dx}\) in terms of \(x\) and \(y\). The equation becomes: \[ \frac{dy}{dx} = \frac{-1 + y e^{xy}}{e^{xy} x} \]

Step 4: Determine Points of Undefined Slope

For the tangent to be vertical, \(x = 0\) would make the denominator zero or when the denominator \(e^{xy} x\) itself is zero. Thus, set the denominator equal to zero and solve: \[e^{xy} x = 0 \rightarrow x = 0\] because \(e^{xy} eq 0\) for real \(xy\).

Step 5: Test Point (0, 1)

Substitute \(x = 0\) into the original equation to check if it satisfies the curve equation. Substitute \(x = 0, y = 1\) into \(y - e^{xy} + x = 0\). Check: \[1 - e^{0 \cdot 1} + 0 = 1 - 1 = 0\], which holds true.

Step 6: Check Other Points (Verification)

Verify the other given points if needed or note that only at \((0, 1)\) the tangent is vertical since no other value satisfies both the condition of undefined slope and the curve equation.

Key Concepts

Implicit Differentiation

Implicit differentiation is a technique used when differentiating equations where the dependent variable cannot be easily isolated. Unlike explicit functions where you can express one variable solely in terms of another, implicit differentiation handles equations involving multiple variables intertwined together.

For example, in the exercise, the curve is represented by the equation \(y - e^{xy} + x = 0\). Here, both \(x\) and \(y\) are mixed in such a way that solving for \(y\) directly in terms of \(x\) isn't straightforward. By differentiating each part of the equation implicitly, we can find \(\frac{dy}{dx}\), the derivative of \(y\) with respect to \(x\):

• The derivative of \(y\) is \(\frac{dy}{dx}\).
• The exponential term \(e^{xy}\) requires using the chain rule, resulting in \(e^{xy} \cdot (y + x \cdot \frac{dy}{dx})\).
• The derivative of \(x\) is 1.

After conducting implicit differentiation on the entire equation, the resulting equation is simplified to find the expression for \(\frac{dy}{dx}\). This approach is invaluable for solving complex derivatives without isolating the function explicitly.

Undefined Slope

An undefined slope typically indicates that a line is vertical. In the context of calculus, this corresponds to where the derivative (\(\frac{dy}{dx}\)) does not exist or where its value is infinite.

For the exercise, a vertical tangent, characterized by undefined slope, is sought on the curve defined by \(y - e^{xy} + x = 0\). To find where \(\frac{dy}{dx}\) becomes undefined, we observe the denominator in the formula for \(\frac{dy}{dx}\): \[\frac{dy}{dx} = \frac{-1 + y e^{xy}}{e^{xy} x}\]The determinant of an undefined slope is when this denominator equals zero. Here, \(e^{xy}x = 0\), the exponential term \(e^{xy}\) never equals zero for real numbers, leaving \(x = 0\) as a potential solution. This insight underlines the point on the curve where our tangent line becomes vertical.

Curve Analysis

Analyzing a curve involves exploring its graphical properties, such as tangents, slopes, and points of interest. In the given problem, we use curve analysis to identify where a vertical tangent exists. This involves confirming the solutions obtained through the derivative for validity on the original curve.

In this exercise, after identifying \(x = 0\) as a condition for a vertical tangent, substitution into the equation \(y - e^{xy} + x = 0\) confirms which values satisfy the equation:

• Substitute \(x = 0\) and test different \(y\)-values.
• Verify if \((x, y)\) points satisfy the equation; only \(y = 1\) succeeds: \(1 - e^{0} + 0 = 0\).

Thus, point \((0, 1)\) is valid on the curve, yielding a vertical tangent. Comprehensive analysis eliminates other given points \((1,1), (1,0), (0,0)\) because they do not satisfy both the derivative condition and the original equation. Such detailed analysis ensures accuracy when determining specific graphic features like vertical tangents on more complex curves.