Question

If (x² - 2xy - y² = 2), then at the point (1, 1):

A. The tangent is horizontal.
B. The tangent is vertical.
C. The tangent is slant.
D. The tangent is undefined.

Answer 1

The slope of the tangent to the curve at the point (1, 1) is determined by taking the partial derivatives of the equation. The partial derivative with respect to x is zero, indicating a horizontal line. Option A is correct: the tangent is horizontal.

Step-by-step explanation:

To determine the nature of the tangent to the curve at the point (1, 1), we must first find the derivative of the given equation. The equation provided is x² - 2xy - y² = 2. To find the slope of the tangent, we need to take the partial derivative with respect to x and y, known as the gradient.

Let's calculate the partial derivatives:

• The partial derivative with respect to x is 2x - 2y.
• The partial derivative with respect to y is -2x - 2y.

At the point (1, 1), the partial derivatives are:

• with respect to x: 2(1) - 2(1) = 0
• with respect to y: -2(1) - 2(1) = -4

Since the partial derivative with respect to x is zero, the slope of the tangent line at the point (1, 1) is also zero. A slope of zero indicates that the tangent line is horizontal at that point. Therefore, the correct option is A: The tangent is horizontal.