Answer:
Undefined.
Step-by-step explanation:
We want to find the slope of the graph of the equation:
[tex]\displaystyle x^2 - y^2 = 81[/tex]
At the point (9, 0).
In other words, we want to evaluate dy/dx when x = 9 and y = 0.
Find dy/dx. We can take the derivative of both sides with respect to x:
[tex]\displaystyle \begin{aligned} \frac{d}{dx}\left[ x^2 - y^2\right] &= \frac{d}{dx}\left [ 81\right] \\ \\ 2x - 2y \frac{dy}{dx} &= 0 \\ \\ \frac{dy}{dx} &= \frac{x}{y}\end{aligned}[/tex]
Then the slope of the graph at the point (9, 0) will be:
[tex]\displaystyle \begin{aligned} \frac{dy}{dx}\Big|_{(9, 0)} &= \frac{(9)}{(0)} \\ \\ &= \text{Und.}\end{aligned}[/tex]
In conclusion, the slope of the graph at the point (9, 0) is undefined.
