Problem

Fix a real number m. Let S1 denote the family of circles, centered on the line y = mx, e...

Fix a real number m. Let S₁ denote the family of circles, centered on the line y = mx, each member of which passes through the origin.

(a) Show that the equation of S₁ can be written in the form where a is a constant that labels particular members of the family.

(b) Determine the equation of the family of orthogonal trajectories to S₁, and show that it consists of the family of circles centered on the line x = −my that pass through the origin.

(c) .

Let F₁ and F₂ be two families of curves with the property that whenever a curve from the family F₁ intersects one from the family F₂, it does so at an angle . If we know the equation of F₂, then it can be shown (see Problem 26 in Section 1.1) that the differential equation for determining F₁ is

where m₂ denotes the slope of the family F₂ at the point (x, y).

Reference : Any curve with the property that whenever it intersects a curve of a given family it does so at an angle

[Hint: From Figure 1.1.6, tan a₁ = tan(a₂ −a). Thus, the equation of the family of oblique trajectories is obtained by solving