Isogonal curves Let a curve be described by r = f(?). where f(?) > 0 on its domain. Referring to the figure of Exercise 50, a curve is isogonal provided the angle ? is constant for all ?.a. Prove that ? is constant for all ? provided cot ? = f?(?)/f(?) is constant, which implies that , where k is a constant.b. Use part (a) to prove that the family of logarithmic spirals r = Cek? consists of isogonal curves, where C and k are constants.c. Graph the curve r = 2e2? and confirm the result of part (b).Exercise 50Tangents and normals Let a polar curve be described by r = f(?) and let ? be the line tangent to the curve at the point P(x, y) = P(r, ?) (see figure).a. Explain why tan ? = dy/dx.b. Explain why tan ? = y/x.c. Let ? be the angle between ? and OP. Prove that tan ? = f(?)/f?(?).d. Prove that the values of ? for which ? is parallel to the x-axis satisfy tan ? = ?f(?)/f?(?).e. Prove that the values of ? for which ? is parallel to the y-axis satisfy tan ? = f?(?)/f(?).

Solution 51AEStep 1:Given thatLet a curve be described by r = f(). where f() > 0 on its domain. Referring to the figure of Exercise 50, a curve is isogonal provided the angle is constant for all .Step 2:To finda. Prove that is constant for all provided cot = f()/f() is constant, which implies that , where k is a constant.b. Use part (a) to prove that the family of logarithmic spirals r = Cek consists of isogonal curves, where C and k are constants.c. Graph the curve r = 2e2 and confirm the result of part (b).Step 3:a. Prove that is constant for all provided cot = f()/f() is constant, which implies that , where k is a constant.Given curve is isogonal curve.That is constant implies is constant=constantNow== constantTherefore, = constant