Determine the points on the negative unit circle where the tangent line is vertical

The negative unit circle is described by the equation:

$ x^2 + y^2 = -1 $

To find where the tangent line is vertical, we need to find the points where the derivative of $ y $ with respect to $ x $ is undefined. First, implicitly differentiate the equation:

$ 2x + 2y \x0crac{dy}{dx} = 0 $

Solving for $ \x0crac{dy}{dx} $:

$ \x0crac{dy}{dx} = -\x0crac{x}{y} $

The derivative is undefined when $ y = 0 $. Substituting $ y = 0 $ into the original equation:

$ x^2 = -1 $

This has no real solutions. Therefore, there are no points on the negative unit circle where the tangent line is vertical.

The negative unit circle is described by:

$ x^2 + y^2 = -1 $

Vertical tangents occur where $ x0crac{dy}{dx} $ is undefined. Implicitly differentiate:

$ 2x + 2y x0crac{dy}{dx} = 0 $

Solving:

$ x0crac{dy}{dx} = -x0crac{x}{y} $

Undefined when $ y = 0 $. Substituting $ y = 0 $:

$ x^2 = -1 $

No real solutions. Thus, no vertical tangents exist.

The negative unit circle is:

$ x^2 + y^2 = -1 $

Find vertical tangents by differentiating:

$ 2x + 2y x0crac{dy}{dx} = 0 $

Solve:

$ x0crac{dy}{dx} = -x0crac{x}{y} $

Undefined when $ y = 0 $. Substitute:

$ x^2 = -1 $

No real solutions, so no vertical tangents.

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