Area enclosed by the curve `y=f(x)` defined parametrically as `x=(1-t^2)/(1+t^2), y=(2t)/(1+t^2)` is equal

Let y=f(x) be defined parametrically as y=t^2+t|t|, x=2t-|t|, t in R , Discuss its continuity.

A function is reprersented parametrically by the equations x=(1+t)/(t^3); y=3/(2t^2)+2/t then the value of |(dy)/(dx)-x((dy)/(dx))^3| is__________

Consider a function y = f(x) defined parametrically as y=2t+t, x=t^2 foralltin R then the function f(x) is

Find (dy)/(dx) when : tan y=(2t)/(1-t^(2)), sin x=(2t)/(1+t^(2))

If a function is represented parametrically be the equations x=(1+(log)_e t)/(t^2); y=(3+2(log)_e t)/t , then which of the following statements are true?

If x=(2t)/(1+t^2) , y=(1-t^2)/(1+t^2) , then find (dy)/(dx) .

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

If t be a variable parameter, show that the point x = a(1 -t^(2))/(1+t^(2)),y = b(2t)/(1 + t^(2)) always lies on an ellipse .

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

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