) and

) and Fig. . P ( x,y ) M O cos , sin , θ π are the parametric equations of the circle x Conversely, if x = a cos , sin , then, x a = cos , a = Squaring and adding, we get, = cos = . Thus x = a yields the equation to circle with centre ( , ) and radius a units. Note ( ) x t y cos , sin , ≤≤ π also represents the same parametric equations of circle x t increasing in anticlockwise direction. ( ) x t y ≤≤ sin , cos , π also represents the same parametric equations of circle x t increasing in clockwise direction. (ii) Parametric form of the parabola y ax Let P x y ( , be a point on the parabola y = ax )( = ( )( = = ( −∞< < ∞ say y = at y t x = at t ( ) x = at Parametric form of y ax is x at at −∞< < ∞ Conversely if x at and y at −∞< < ∞ , then eliminating ' ' t between these equations we get y ax (iii) Parametric form of the Ellipse x Let P be any point on the ellipse. Let the ordinate MP meet the auxiliary circle at Q . Let ∠ ACQ = α ∴ CM = a MQ cos , and Q a ( cos , sin Now x -coordinate of P is a cos α . If its y -coordinate is ′ y , then P a ( cos , ′ lies on = Fig. . Fig. . Fig. . Q M C P ′ Two Dimensional Analytical Geometry - II whence cos α + ′ y = ⇒ ′ y = b sin α . Hence P is ( cos , sin α . The parameter α is called the eccentric angle of the point P . Note that α is the angle which the line CQ makes with the x -axis and not the angle which the line CP makes with it. Hence the parametric equation of an ellipse is x cos , θ , where θ is the parameter π . (iv) Parametric form of the Hyperbola x Similarly, parametric equation of a hyperbola can be derived as x sec , θ , where θ is the parameter. − π except θ = ± . In nutshell the parametric equations of the circle, parabola,ellipse and hyperbola are given in the following table. Conic Parametric equations Parameter Range of parameter Any point on the conic Circle cos θ sin θ ‘ θ ’ or ( cos , sin ) Parabola at at = −∞< < ∞ ‘ t ’ or at at Ellipse cos θ sin θ ‘ θ ’ or ( cos , sin ) Hyperbola sec θ tan θ except θ = ± ‘ θ ’ or ( sec , tan ) Remark ( ) Parametric form represents a family of points on the conic which is the role of a parameter. Further parameter plays the role of a constant and a variable, while cartesian form represents the locus of a point describing the conic. Parameterisation denotes the orientation of the curve. ( ) A parametric representation need not be unique. ( ) Note that using parameterisation reduces the number of variables at least by one.