91Ó°ÊÓ

The derivative is a crucial concept in calculus that helps us understand how a function changes at any given point. It tells us the rate of change, or the slope, of the tangent line to the curve at a specific point. In this exercise, the derivatives of the given curves are found to determine the slopes of their tangent lines.
For the family of circles with the equation \( x^2 + y^2 = r^2 \), we differentiate it with respect to \( x \) to find the slope of the tangent. The derivative \( \frac{dy}{dx} = -\frac{x}{y} \) gives us the slope at any point on the circle. Similarly, for the line \( ax + by = 0 \), differentiating gives us \( \frac{dy}{dx} = -\frac{a}{b} \), representing the slope of the tangent to the line.
Understanding these derivatives helps us evaluate the relationship between the two curves by examining their slopes, which is key to finding orthogonal trajectories.

Tangent lines touch curves at exactly one point and are essential for studying the behavior of curves at that specific point. They provide crucial information about the direction and steepness of the curve where they meet.
In our exercise, the slope of the tangent to the circle is \( -\frac{x}{y} \), which varies depending on the position on the circle. For the line, the slope is constant, \( -\frac{a}{b} \).

• Tangent to the circle: showcases how the slope varies across different points.
• Tangent to the line: offers a consistent slope across its entire length.

These tangents and their slopes are fundamental to analyzing how curves can intersect each other at right angles.

Intersection points of curves are where the equations of the curves meet. At these points, the tangent lines are particularly important for determining the interaction between curves. In mathematical terms, these points allow us to study the properties like orthogonality.
Here, the given curves, circles, and straight lines intersect at the origin and potentially other points determined by their characteristics. To analyze their intersection, we calculate the product of the slopes of their tangents. If these products equal \(-1\) at any intersection, then the curves are orthogonal.
This intersection concept is crucial because it helps verify if one family of curves is indeed orthogonal to another, which is a significant relationship in the world of curve analysis.

Orthogonality is a special property where two curves intersect at right angles. In cartesian geometry, this property is used to show that curves are mutually perpendicular, making them orthogonal trajectories of each other.
For the given exercise, orthogonality is verified by ensuring that the product of the slopes of the tangent lines is precisely \(-1\) at their intersection points.

• This means multiplying the tangent slope of the circle \( -\frac{x}{y} \) by the tangent slope of the line \( -\frac{a}{b} \) gives \( \frac{ax}{by} = -1 \).
• As the conditions are met, the families \( x^2 + y^2 = r^2 \) and \( ax + by = 0 \) are indeed orthogonal trajectories.

Understanding such orthogonal interactions helps in visualizing and sketching how these curves are placed on a coordinate system, providing a powerful tool for graphical analysis.