Consider the system of hyperbola `xy = k, k in R`. Let `e_1` be the eccentricity when k = 4 and `e_2` be the eccentricity when k = 9, then `e_1-e_2`

Consider an ellipse x^2/a^2+y^2/b^2=1 Let a hyperbola is having its vertices at the extremities of minor axis of an ellipse and length of major axis of an ellipse is equal to the distance between the foci of hyperbola. Let e_1 and e_2 be the eccentricities of an ellipse and hyperbola respectively. Again let A be the area of the quadrilateral formed by joining all the foci and A, be the area of the quadrilateral formed by all the directrices. The relation between e_1 and e_2 is given by

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 .

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 .

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_1 is the eccentricity of the conic 9x^2+4y^2=36\ a n d\ e_2 is the eccentricity of the conic 9x^2-4y^2=36 then a. e_1 ^2-e_2^ 2=2 b. e_2 ^2-e_1 ^2=2 c. 2 3

Let F_1,F_2 are the foci of the hyperbola x^2/16-y^2/9=1 and F_3,F_4 are the foci of its conjugate hyperbola. If e_H and e_C are their eccentricities respectivley then the statement which holds true is:

A series of concentric ellipses E_1,E_2, E_3..., E_n are drawn such that E touches the extremities of the major axis of E_(n-1) , and the foci of E_n coincide with the extremities of minor axis of E_(n-1) If the eccentricity of the ellipses is independent of n, then the value of the eccentricity, is

The equation of directrix of a hyperbola is 3x + 4y + 8 = 0 The focus of the hyperbola is (1, 1) . If eccentricity of the hyperbola is 2 and the length of conjugate axis is k then [k], where [] represents the greatest integer function, is equal to ________ .

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 .

The latus rectum of a hyperbola (x^(2))/( 16) -(y^(2))/( p) =1 is 4(1)/(2) .Its eccentricity e=