The families of curves defined by the equations y=a x, y^2+x^2=c^2 are perpendicular for (a) a=2

step 1 In this problem, for the family of the curves to be perpendicular to each other means the slope,

The families of curves defined by the equations y=a x,...

Key Concepts

Families of Curves

In mathematics, a family of curves is a collection of curves that are defined by an equation containing one or more parameters. These parameters can be varied to generate different members of the family, and analyzing how the curves change with the parameters is important in understanding the overall behavior of the system. In many problems, families of curves are used to study common geometric or analytic properties shared by the curves in the family.

Orthogonal Trajectories

Orthogonal trajectories refer to a family of curves that intersect another family of curves at right angles (90°). When two curves are orthogonal at their intersection points, the product of the slopes of their tangents at that point is -1. This concept is fundamental in areas such as differential equations and geometry, where finding a family of curves that intersect another family perpendicularly can lead to deeper insights into the underlying system or physical phenomena.

Slope Criterion for Perpendicularity

The slope criterion for perpendicularity is based on the fact that two lines are perpendicular if and only if the product of their slopes is -1. In the context of curves, one considers the slopes of their tangent lines at the point of intersection. This condition is used to determine whether or not two curves intersect orthogonally, and it is a critical tool in problems that mix algebra and differential calculus.

Tangent Lines to Curves

Tangent lines provide the best linear approximation to a curve at a given point, and their slopes are determined by taking the derivative of the curve’s equation with respect to the independent variable. When analyzing intersections of curves, particularly in determining orthogonality, calculating the tangent slopes is essential. This concept connects differential calculus with geometric interpretations of curves and their interactions.

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