) 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