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

Chapter 1, Problem 32

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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 theform(x a)2 + (y ma)2 = a2(m2 + 1),where a is a constant that labels particular membersof the family.(b) Determine the equation of the family of orthogonaltrajectories to S1, and show that it consistsof the family of circles centered on the linex = my that pass through the origin.(c) Sketch some curves from both families whenm = 3/3.Let F1 and F2 be two families of curves with the propertythat whenever a curve from the family F1 intersectsone from the family F2, it does so at an anglea = /2. If we know the equation of F2, then it canbe shown (see in Section 1.1) that the differentialequation for determining F1 isdydx = m2 tan a1 + m2 tan a, (1.8.16)where m2 denotes the slope of the family F2 at thepoint (x, y).