Verify that the following pairs of curves meet orthogonally....

Verify that the following pairs of curves meet orthogonally.
\begin{equation}
\begin{array}{l}{\text { a. } x^{2}+y^{2}=4, \quad x^{2}=3 y^{2}} \\ {\text { b. } x=1-y^{2}, \quad x=\frac{1}{3} y^{2}}\end{array}
\end{equation}

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Key Concepts

Slope of Tangent Lines

The slope of a tangent line to a curve at a given point is found by differentiating the curve's equation to obtain dy/dx. This derivative represents the instantaneous rate of change of the curve and is crucial in analyzing how curves meet, especially when verifying conditions such as perpendicularity.

Intersection of Curves

The intersection of curves involves finding common points that satisfy the equations of both curves simultaneously. Determining these intersection points is essential before evaluating the slopes of the tangents at those points to check for orthogonality.

Orthogonal Curves

Orthogonal curves are curves that intersect at right angles. This means that at each point of intersection, the slopes of their tangents are negative reciprocals of each other, i.e., the product of the slopes is -1. This property is often used to verify if two curves intersect orthogonally.

Implicit Differentiation

Implicit differentiation is a technique used to differentiate equations that are not explicitly solved for one variable in terms of another. It is particularly useful when dealing with curves defined by relations where y cannot be easily isolated, and it allows us to calculate the slope of a curve at any point.

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