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

If a variable line has its intercepts on the coordinate axes ea n de^(prime), where e/2 and e^,/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

Statement-I (5)/(3) and (5)/(4) are the eccentricities of two conjugate hyperbolas. Statement-II If e_1 and e_2 are the eccentricities of two conjugate hyperbolas, then e_1e_2gt1 .

If e and e_(1) , are the eccentricities of the hyperbolas xy=c^(2) and x^(2)-y^(2)=c^(2) , then e^(2)+e_(1)^(2) is equal to

The ellipse (x^(2))/(a^(2))+(y^(2))/(b^(2))=1 and the hyperbola (x^(2))/(A^(2))-(y^(2))/(B^(2))=1 are given to be confocal and length of mirror axis of the ellipse is same as the conjugate axis of the hyperbola. If e_1 and e_2 represents the eccentricities of ellipse and hyperbola respectively, then the value of e_(1)^(-2)+e_(1)^(-2) is

Statement-I If eccentricity of a hyperbola is 2, then eccentricity of its conjugate hyperbola is (2)/(sqrt(3)) . Statement-II if e and e_1 are the eccentricities of two conjugate hyperbolas, then ee_1gt1 .

Let the major axis of a standard ellipse equals the transverse axis of a standard hyperbola and their director circles have radius equal to 2R and R respectively. If e_(1) and e_(2) , are the eccentricities of the ellipse and hyperbola then the correct relation is

If radii of director circle of the ellipse (x^(2))/(a^(2))+(y^(2))/(b^(2))=1 and hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 are in the ratio 1:3 and 4e_(1)^(2)-e_(2)^(2)=lambda , where e_1 and e_2 are the eccetricities of ellipse and hyperbola respectively, then the value of lambda is

If e and e' are the eccentricities of the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2)) =1 and (y^(2))/(b^(2))-(x^(2))/(a^(2))=1 , then the point ((1)/(e),(1)/(e')) lies on the circle:

If e and e' are the eccentricities of the ellipse 5x^(2) + 9 y^(2) = 45 and the hyperbola 5x^(2) - 4y^(2) = 45 respectively , then ee' is equal to

Let E1 and E2, be two ellipses whose centers are at the origin.The major axes of E1 and E2, lie along the x-axis and the y-axis, respectively. Let S be the circle x^2 + (y-1)^2= 2 . The straight line x+ y =3 touches the curves S, E1 and E2 at P,Q and R, respectively. Suppose that PQ=PR=[2sqrt2]/3 .If e1 and e2 are the eccentricities of E1 and E2, respectively, then the correct expression(s) is(are):