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

If t is a parameter, then x=a(t+(1)/(t)),y=b(t-(1)/(t)) represents

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

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

The slope of the tangent to the curve x= 3t^(2) +1 , y=t^(3)-1 at x=1 is-

The locus of the moving point whose coordinates are given by (e^t+e^(-t),e^t-e^(-t)) where t is a parameter, is x y=1 (b) x+y=2 x^2-y^2=4 (d) x^2-y^2=2

The locus of the moving point whose coordinates are given by (e^t-e^(-t),e^t+e^(-t)) where t is a parameter, is (a) x y=1 (b) y+x=2 (c) y^2-x^2=4 (d) y^2-x^2=2

If the tangent at a point P, with parameter t, on the curve x=4t^(2)+3,y=8t^(3)-1,t in RR , meets the curve again at a point Q, then the coordinates of Q are

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, 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) .

If t is a variable parameter, show that x=(a)/(2)(t+(1)/(t)) and y = (b)/(2)(t-(1)/(t)) represent a hyperbola.