In the curve represented parametrically by the equations `x=2logcott+1` and `y=tant+cott ,`

If a curve is represented parametrically by the equation x=f(t) and y=g(t)" then prove that "(d^(2)y)/(dx^(2))=-[(g'(t))/(f'(t))]^(3)((d^(2)x)/(dy^(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?

A curve is defined parametrically by the the equation x=t^(2) and y=t^(3). A variable pair of peerpendicular lines through the origin O meet the curve at P and Q. If the locus of the point of intersectin of the tangents at P and Q is ay^(2)=bx-1, then the value of (a+b) is "______."

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

If x and y are connected parametrically by the equations given in Exercises 1 to 10, without eliminating the parameter, Find (dy)/(dx) . x= 2at, y= at^(4) .

If x and y are connected parametrically by the equations given in Exercises 1 to 10, without eliminating the parameter, Find (dy)/(dx) . x= a(cos t+ log tan (t/2)), y= a sin t .

If x and y are connected parametrically by the equations given in Exercises 1 to 10, without eliminating the parameter, Find (dy)/(dx) . x = sin t, y= cos 2t .

Find the centre of the circle whose parametric equation is x= -1+2 cos theta , y= 3+2 sin theta .

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