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 locus represented by x=(1)/(2)a(t+(1)/(t)),y=(1)/(2)a(t-(1)/(t)) is a rectangular hyperbola.

Show that the locus represented by x=(1)/(2)a(t+(1)/(t)),y=(1)/(2)a(t-(1)/(t)) is a rectangular hyperbola.

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 the locus of the point ((a)/(2)(t+(1)/(t)),(a)/(2)(t-(1)/(t))) represents a conic,then distance between the directrices is

The locus of the point x=(t^(2)-1)/(t^(2)+1),y=(2t)/(t^(2)+1)

The locus of a point represent by x=(a)/(2)((t+1)/(t)),y=(a)/(2)((t-1)/(t)) , where t=in R-{0} , is

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) .

Show that the locus represented by x = 1/2 a (t + 1/t) , y = 1/2 a (t - 1/t) is a rectangular hyperbola. Show also that equation to the normal at the point 't' is x/(t^(2) + 1) + y/(t^(2) - 1) = a/t .

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

The equation x =1/2 (t+ (1)/(t)), y = 1/2 (t - 1/t), t ne 0 represents