 A student claims that the ellipse x2 −xy+y2 =1 has a horizontal tangent line at the point (1,1). Without doing any computations, explain why the student’s claim must be incorrect

The student's claim that the ellipse \(x^2 - xy + y^2 = 1\) has a horizontal tangent line at the point \((1,1)\) must be incorrect. Here's why:

For a curve to have a horizontal tangent line at a given point, the slope of the tangent line (which is given by the derivative \(\frac{dy}{dx}\)) must be zero at that point.

However, let's examine the point \((1,1)\) on the given curve \(x^2 - xy + y^2 = 1\). Notice that this curve is symmetric with respect to the line \(y = x\). The symmetry implies that if \((1,1)\) lies on the curve, then the curve should behave similarly in the directions of both \(x\) and \(y\) around this point.

At the point \((1,1)\), both \(x\) and \(y\) are equal. For a horizontal tangent to exist at \((1,1)\), the slope of the curve at this point should be zero. But due to the symmetry of the equation with respect to \(x\) and \(y\), the slope is more likely to be either positive or negative, depending on the interaction of the \(x\) and \(y\) terms.

Specifically, the term \(-xy\) in the equation introduces a coupling between \(x\) and \(y\), which means the slope at \((1,1)\) is unlikely to be zero because both partial derivatives of the equation with respect to \(x\) and \(y\) will generally contribute non-zero terms that don't cancel out symmetrically.

Thus, without doing any computations, the symmetry and the nature of the equation suggest that the slope is not zero at \((1,1)\), so there cannot be a horizontal tangent line at that point. Therefore, the student's claim must be incorrect.

Would you like details or have any questions?

Here are 8 questions to expand on this information:

1. How can we formally compute the slope of the tangent line to verify the claim?
2. What does it mean for a curve to have a tangent line at a specific point?
3. How can we find the equation of the tangent line to a curve at a given point?
4. Why is symmetry important when analyzing curves?
5. What role do partial derivatives play in determining the slope of a tangent line?
6. How can we check if a curve is symmetric with respect to the line \(y = x\)?
7. What other methods can be used to determine the nature of the tangent line at a given point?
8. How would the analysis change if the curve were given in a different form?

**Tip:** When analyzing the slope of a tangent line, always consider the symmetry and interaction of the terms in the equation before performing computations.

Explore why an ellipse defined by the equation x^2 - xy + y^2 = 1 cannot have a horizontal tangent line at the point (1,1). Learn about ellipse symmetry, the concept of tangent lines, and why the slope of the tangent line cannot be zero at this specific point. Suitable for advanced high school students and beyond.

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