Chapter 1 Topics in Analytic Geometry

MA112 Section 750001: Prepared by Dr.Archara Pacheenburawana 23Symmetry TestsObserve that the polar graph r = cos2θ in above Figure is symmetric about the x-axis andthe y-axis. This symmetry could have been predicted from the following theorem.Theorem 2.1 (Symmetry Tests).(a) A curve in polar coordinates is symmetric about the x-axis if replacing θ by −θ in itsequation produces an equivalent equation.(b) A curve in polar coordinates is symmetric about the y-axis if replacing θ by π −θ inits equation produces an equivalent equation.(c) A curve in polar coordinates is symmetric about the origin if replacing θ by θ+π, orreplacing r by −r in its equation produces an equivalent equation.π/2(r,θ)(r,π −θ)π/2(r,θ)π/2(r,θ)000(r,−θ)(r,θ+π)or(−r,θ)(a)(b)(c)Example 2.15 Show that the graph of r = cos2θ is symmetric about the x-axis and y-axis.Solution .........Families of Lines and Rays Through the PoleFor any constant θ 0 , the equationθ = θ 0 (2.6)is satisfied by the coordinates of the form P(r,θ 0 ), regardless of the value of r. Thus, theequation represents the line that passes through the pole and makes an angle of θ 0 with thepolar axis.π/2θ0θ = θ 0