Question

Provide an appropiate response.

At the two points where the curve x² + 2xy + y² = 9 crosses the x-axis, the tangents to the curve are parallel. What is the common slope of these tangents?

Answer 1

-1

Explanation:

We can find the tangents to the curve at x = 0, where the roots of the function occur (function crosses the x-axis). We can differentiate the function with respect to x by using implicit differentiation.

• 
  `\displaystyle (d)/(dx) \Big [ x^2 + 2xy + y^2 = 9 \Big ]`

We will need to use the Power Rule and Product Rule to differentiate this function.

• 
  `\displaystyle 2x + 2 \Big ( x \cdot (dy)/(dx) + y \cdot 1 \Big ) + 2y \cdot (dy)/(dx) = 0`

Distribute 2 inside the parentheses.

• 
  `\displaystyle 2x + 2x (dy)/(dx) + 2y + 2y (dy)/(dx) = 0`

Move all terms containing dy/dx to the left side of the equation, and all other terms to the right side of the equation.

• 
  `\displaystyle 2x (dy)/(dx) + 2y (dy)/(dx) = -2x-2y`

Factor dy/dx from the left side of the equation.

• 
  `\displaystyle (dy)/(dx) \Big (2x + 2y \Big ) = -2x-2y`

Divide both sides of the equation by 2x + 2y.

• 
  `\displaystyle (dy)/(dx) = (-2x-2y)/(2x+2y)`

Factor the negative sign out of the numerator.

• 
  `\displaystyle (dy)/(dx) = -(2x+2y)/(2x+2y)`

Cancel out the numerator and the denominator.

• 
  `\displaystyle (dy)/(dx) = -1`

The derivative of the function is -1 at all points, meaning that everywhere the function has a slope, or tangent, of -1.

The common slope of the tangents as the curve x² + 2xy + y² = 9 crosses the x-axis is -1.