The vertex of the parabola whose parametric equation is `x=t^2-t+1,y=t^2+t+1; t in R ,` is

If t be a variable parameter, find the vartex, axis, focus and length of the latus rectum of the parabola whose parametric equations are, x = u cos alpha* t, y = u sin alpha * t - (1)/(2) "gt"^(2) .

Find the equation to the locus represented by the parametric equations x=2t^(2)+t+1, y=t^(2)-t+1 .

A curve in the xy-plane is parametrically given by x=t+t^3a n dy=t^2,w h e r et in R is the parameter. For what value(s) of t is (dy)/(dx)=1/2? 1/3 b. 2 c. 3 d. 1

The derivative of the function represented parametrically as x=2t=|t|,y=t^3+t^2|t|a tt=0 is a. -1 b. 1 c. 0 d. does not exist

Find the equation of the hyperbola given by equations x=(e^t+e^(-t))/2 and y=(e^t-e^(-t))/3,t in R .

Show that the curve whose parametric coordinates are x=t^(2)+t+l,y=t^(2)-t+1 represents a parabola.

The parametric equation of a parabola is x=t^2-1,y=3t+1. Then find the equation of the directrix.

Computing area with parametrically represented boundaries : If the boundary of a figure is represented by parametric equation, i.e., x=x(t), y=(t), then the area of the figure is evaluated by one of the three formulas : S=-int_(alpha)^(beta) y(t)x'(t)dt, S=int_(alpha)^(beta) x(t)y'(t)dt, S=(1)/(2)int_(alpha)^(beta)(xy'-yx')dt, Where alpha and beta are the values of the parameter t corresponding respectively to the beginning and the end of the traversal of the curve corresponding to increasing t. If the curve given by parametric equation x=t-t^(3), y=1-t^(4) forms a loop for all values of t in [-1,1] then the area of the loop is

The parametric equation of a parabola is x=t^2+1,y=2t+1. Then find the equation of the directrix.

Computing area with parametrically represented boundaries : If the boundary of a figure is represented by parametric equation, i.e., x=x(t), y=(t), then the area of the figure is evaluated by one of the three formulas : S=-int_(alpha)^(beta)y(t)x'(t)dt, S=int_(alpha)^(beta)x(t)y'(t)dt, S=(1)/(2)int_(alpha)^(beta)(xy'-yx')dt, Where alpha and beta are the values of the parameter t corresponding respectively to the beginning and the end of the traversal of the curve corresponding to increasing t. The area of the loop described as x=(t)/(3)(6-t),y=(t^(2))/(8)(6-t) is