Find the eccentricity of the hyperbola given by equations `x=(e^t+e^(-1))/2a n dy=(e^t-e^(-1))/3,t in Rdot`

For each of the differential equations in (e^(x)+e^(-x))dy-(e^(x)-e^(-x))dx=0

If e_1 is the eccentricity of the ellipse x^2/16+y^2/25=1 and e_2 is the eccentricity of the hyperbola passing through the foci of the ellipse and e_1 e_2=1 , then equation of the hyperbola is

If e_(1) is the eccentricity of the ellipse (x^(2))/(25)+(y^(2))/9=1 and if e_(2) is the eccentricity of the hyperbola 9x^(2)-16y^(2)=144 , then e_(1)e_(2) is. . . . .

An ellipse and a hyperbola are confocal (have the same focus) and the conjugate axis of the hyperbola is equal to the minor axis of the ellipse. If e_1a n de_2 are the eccentricities of the ellipse and the hyperbola, respectively, then prove that 1/(e1 2)+1/(e2 2)=2 .

Find the particular solution of the differential equation (1+e^(2x))dy+(1+y^(2))e^(x)dx=0. Given that y=1 when x=0.

Integrate the rational functions 1/((e^(x)-1)) [Hint : Put e^(x) = t]

A curve is represented parametrically by the equations x=t+e^(at) and y=-t+e^(at) when t in R and a > 0. If the curve touches the axis of x at the point A, then the coordinates of the point A are

If |cos^(-1)((1-x^2)/(1+x^2))|

Find the locus of the point (t^2-t+1,t^2+t+1),t in Rdot