Answer:
-1
Step-by-step 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.
- [tex]\displaystyle \frac{d}{dx} \Big [ x^2 + 2xy + y^2 = 9 \Big ][/tex]
We will need to use the Power Rule and Product Rule to differentiate this function.
- [tex]\displaystyle 2x + 2 \Big ( x \cdot \frac{dy}{dx} + y \cdot 1 \Big ) + 2y \cdot \frac{dy}{dx} = 0[/tex]
Distribute 2 inside the parentheses.
- [tex]\displaystyle 2x + 2x \frac{dy}{dx} + 2y + 2y \frac{dy}{dx} = 0[/tex]
Move all terms containing dy/dx to the left side of the equation, and all other terms to the right side of the equation.
- [tex]\displaystyle 2x \frac{dy}{dx} + 2y \frac{dy}{dx} = -2x-2y[/tex]
Factor dy/dx from the left side of the equation.
- [tex]\displaystyle \frac{dy}{dx} \Big (2x + 2y \Big ) = -2x-2y[/tex]
Divide both sides of the equation by 2x + 2y.
- [tex]\displaystyle \frac{dy}{dx} = \frac{-2x-2y}{2x+2y}[/tex]
Factor the negative sign out of the numerator.
- [tex]\displaystyle \frac{dy}{dx} = -\frac{2x+2y}{2x+2y}[/tex]
Cancel out the numerator and the denominator.
- [tex]\displaystyle \frac{dy}{dx} = -1[/tex]
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.
