Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
b. On every ellipse, there are exactly two points at which the curve has slope s, where s is any real number.

Verified step by step guidance

1. Recall that an ellipse can be represented by the equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), where \(a\) and \(b\) are positive constants.
2. To find the slope of the tangent line at any point on the ellipse, implicitly differentiate the ellipse equation with respect to \(x\): \(\frac{d}{dx} \left( \frac{x^2}{a^2} + \frac{y^2}{b^2} \right) = \frac{d}{dx} (1)\).
3. Applying the derivative, we get: \(\frac{2x}{a^2} + \frac{2y}{b^2} \frac{dy}{dx} = 0\). Solve for \(\frac{dy}{dx}\) (the slope \(s\)): \(\frac{dy}{dx} = s = -\frac{b^2}{a^2} \frac{x}{y}\).
4. Notice that the slope depends on both \(x\) and \(y\). For a fixed slope \(s\), rearrange the equation to express \(y\) in terms of \(x\) and \(s\): \(s = -\frac{b^2}{a^2} \frac{x}{y} \implies y = -\frac{b^2}{a^2} \frac{x}{s}\) (assuming \(s \neq 0\)).
5. Substitute this expression for \(y\) back into the ellipse equation to find the \(x\)-coordinates where the slope is \(s\). This will lead to a quadratic equation in \(x\), which can have zero, one, or two real solutions depending on \(s\). Therefore, the number of points on the ellipse with slope \(s\) can vary, and it is not always exactly two.

Key Concepts

Slope of a Curve and Tangent Lines

The slope of a curve at a point is the derivative of the function defining the curve at that point. It represents the slope of the tangent line to the curve, indicating the instantaneous rate of change. Understanding how to find and interpret slopes is essential for analyzing points where the curve has a specific slope.

Parametric and Implicit Differentiation of Ellipses

Ellipses are often defined implicitly or parametrically, requiring implicit differentiation to find the slope of the tangent line. This process involves differentiating both sides of the ellipse equation with respect to x or a parameter, allowing determination of slope values at various points on the ellipse.

Number of Solutions to Slope Equations on Ellipses

Determining how many points on an ellipse have a given slope involves solving an equation derived from the derivative equal to that slope. The number of solutions depends on the ellipse's shape and the slope value, and it is not always exactly two; some slopes may correspond to one, two, or no points, which is key to evaluating the statement.