Prove that the point `{(a)/(2)(t+(1)/(t)), (b)/(2)(t-(1)/(t))}` lies on the hyperbola for all values of `t(tne0)`.

Show that the point : x=(2rt)/(1+t^2), y=(r(1-t^2))/(1+t^2) (r constant) lies on a circle for all values of t such that -1 le t le 1 .

Show that for all values of t the point x = a. (1 + t^(2))/(1-t^(2)), y = (2at)/(1-t^(2)) lies on a fixed hyperbola. What is the value of the eccentricity of the hyperbola?

Show that the point (x ,y) given by x=(2a t)/(1+t^2)a n dy=((1-t^2)/(1+t^2)) lies on a circle for all real values of t such that -1lt=tlt=1, where a is any given real number.

Show that the point (x ,y) given by x=(2a t)/(1+t^2)a n dy=((1-t^2)/(1+t^2)) lies on a circle for all real values of t such that -1lt=tlt=1, where a is any given real number.

Show that the point (x,y) given by x=(2at)/(1+t^(2)) and y=((1-t^(2))/(1+t^(2))) lies on a circle for all real values of t such that -1<=t<=1 where a is any given real number.

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

The eccentricity of the hyperbola x= ( a)/(2) ( t+1//t), y=(a)/(2) (t-1//t) is

The locus of the point ( (e^(t) +e^(-t))/( 2),(e^t-e^(-t))/(2)) is a hyperbola of eccentricity

The locus of the point ( (e^(t) +e^(-t))/( 2),(e^t-e^(-t))/(2)) is a hyperbola of eccentricity

For any real t,x=(1)/(2)(e^(t)+e^(-t)),y=(1)/(2)(e^(t)-e^(-t)) is a point on the hyperbola x^(2)-y^(2)=1 show that the area bouyped by the hyperbola and the lines joining its centre to the points corresponding to t_(1)and-t_(1) ist _(1) .