Chapter 4.2, Problem 16E

To determine

To define: The slope of the given curve.

Answer to Problem 16E

Slope is defined for all values of $x\text{and}y$ except when $x=2y$ .

Explanation of Solution

Given information: Given curve $x^2+4xy+4y^2-3x=6$ .

Formula used: $\frac{d(x^n)}{dx}=n{x}^{n-1}\text{and}\frac{d(\text{constant})}{dx}=0$ with product, chain, subtraction and addition rule.

Calculation:

Let the intersection points of the curve and tangent is $P(x,y)$

also, the slope of tangent at given points,

  $\begin{array}{l}{x}^2+4xy+4y^2-3x=6\\ \text{Differentiate}\text{with}\text{respect}\text{to}x\\ \frac{d(x^2)}{dx}+4x\times \frac{dy}{dx}+4\times \frac{d(x)}{dx}\times y+4\times \frac{d(y^2)}{dy}\times \frac{dy}{dx}-3\times \frac{d(x)}{dx}=\frac{d(6)}{dx}\\ 2x+4x\frac{dy}{dx}+4y+4\times 2y\times \frac{dy}{dx}-3=0\\ 4x\frac{dy}{dx}+8y\frac{dy}{dx}=3-2x-4y\\ (4x-8y)\frac{dy}{dx}=3-2x-4y\\ {\frac{dy}{dx}}_{(x,y)}=\frac{3-2x-4y}{4x-8y}\end{array}$

Now, the slope is not defined, when,

  $\begin{array}{l}4x-8y=0\\ x-2y=0\\ x=2y\end{array}$

Thus, slope is defined for all values of $x\text{and}y$ except when $x=2y$ .