If `ea n de '` the eccentricities of a hyperbola and its conjugate, prove that `1/(e^2)+1/(e '^2)=1.`

If a,b are eccentricities of a hyperbola and its conjugate hyperbola then a^(-2)+b^(-2)=

Let e_1 and e_2 be the eccentricities of a hyperbola and its conjugate hyperbola respectively. Statement 1 : e_1 e_2 gt sqrt(2) . Statement 2 : 1/e_1^2+ 1/e_2^2 = ?

If a variable line has its intercepts on the coordinate axes ea n de^(prime), where e/2a n d e^(prime/)2 are the eccentricities of a hyperbola and its conjugate hyperbola, then the line always touches the circle x^2+y^2=r^2, where r= 1 (b) 2 (c) 3 (d) cannot be decided

If a variable line has its intercepts on the coordinate axes ea n de^(prime), where e/2a n d e^(prime)/2 are the eccentricities of a hyperbola and its conjugate hyperbola, then the line always touches the circle x^2+y^2=r^2, where r= (a) 1 (b) 2 (c) 3 (d) cannot be decided

If a variable line has its intercepts on the coordinate axes e and e^(prime), where e/2 and e^(prime)/2 are the eccentricities of a hyperbola and its conjugate hyperbola, then the line always touches the circle x^2+y^2=r^2, where r= (a)1 (b) 2 (c) 3 (d) cannot be decided

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 .

If 5/4 is the eccentricity of a hyperbola find the eccentricity of its conjugate hyperbola.

Let e be the eccentricity of a hyperbola and f(e ) be the eccentricity of its conjugate hyperbola then int_(1)^(3)ubrace(fff...f(e))_("n times") de is equal to

Statement- 1 : If 5//3 is the eccentricity of a hyperbola, then the eccentricity of its conjugate hyperbola is 5//4 . Statement- 2 : If e and e' are the eccentricities of hyperbolas (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 and (x^(2))/(a^(2))-(y^(2))/(b^(2))=-1 respectively, then (1)/(e^(2))+(1)/(e'^(2))=1 .

If the eccentricity of the hyperbola conjugate to the hyperbola (x^2)/(4)-(y^2)/(12)=1 is e, then 3e^2 is equal to: