Question

Find the slope of the graph of the equation at the given point. (If an answer is undefined, enter UNDEFINED.)


`x^(2) -y^(2) =81, (9,0)`

Answer 1

Undefined.

Explanation:

We want to find the slope of the graph of the equation:

`\displaystyle x^2 - y^2 = 81`

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:

`\displaystyle \begin{aligned} (d)/(dx)\left[ x^2 - y^2\right] &= (d)/(dx)\left [ 81\right] \\ \\ 2x - 2y (dy)/(dx) &= 0 \\ \\ (dy)/(dx) &= (x)/(y)\end{aligned}`

Then the slope of the graph at the point (9, 0) will be:

`\displaystyle \begin{aligned} (dy)/(dx)\Big|_((9, 0)) &= ((9))/((0)) \\ \\ &= \text{Und.}\end{aligned}`

In conclusion, the slope of the graph at the point (9, 0) is undefined.