https://file.aiquickdraw.com/mathbot/file/1727968454275gufgt132.jpg

The problem provides the equation of the ellipse:

\[
\frac{x^2}{36} + \frac{y^2}{25} = 1
\]

The slope of the tangent line to the ellipse at any arbitrary point \((x, y)\) is given by the formula:

\[
\text{slope} = \frac{-25x}{36y}
\]

Next, the question asks whether there are points on the ellipse where the slope is undefined. The slope is undefined when the denominator \(36y\) is zero, which happens when \(y = 0\).

**Step 1: Identify Points where Slope is Undefined**
- The slope is undefined when \(y = 0\). 
- To find the points on the ellipse where \(y = 0\), substitute \(y = 0\) into the ellipse equation:

\[
\frac{x^2}{36} + \frac{0^2}{25} = 1
\]
\[
\frac{x^2}{36} = 1
\]
\[
x^2 = 36
\]
\[
x = \pm 6
\]

So, when \(y = 0\), the points on the ellipse are \((6, 0)\) and \((-6, 0)\).

**Answer:**
The points on the ellipse where the slope is undefined are \((6, 0)\) and \((-6, 0)\).

Would you like further details or clarifications?

### Related Questions:
1. What is the geometric significance of the slope being undefined at these points on the ellipse?
2. How would you find the equation of the tangent line at a point where the slope is defined?
3. How does the formula for the slope of the tangent line change if the ellipse has a different equation?
4. Can the same approach be used to find the slope of the tangent line for a hyperbola?
5. How does the slope of the tangent line relate to the derivative of the implicit equation of the ellipse?

### Tip:
For conic sections like ellipses, the tangent line's slope can often be derived from implicit differentiation.

Find the slope of the tangent line to the ellipse x^2/36 + y^2/25 = 1 at an arbitrary point (x, y). Are there points on the ellipse at which the slope is undefined? If so, enter them as comma-separated ordered pairs. If not, enter none.

Learn how to find the slope of the tangent line to the ellipse x^2/36 + y^2/25 = 1 at any arbitrary point (x, y) using the formula -25x/36y. Understand when the slope is undefined and identify points such as (6, 0) and (-6, 0) on the ellipse.

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