A sequence of ellipse `E_1,E_2,E_3,…,E_n` is constructed as foolow: Ellipse `E_n-1` as the extremities of the major axis of `E_n-1` and to have its foci at the extrimities of the mirror axis of `E_n-1`. If eccentricity of ellipse `E_n` is `e_n`, then the locus of `(e_(n)^(2)n,e_(n)^(2)-1` is

A series of concentric ellipse E_1,E_2,E_3,…,E_n is constructed as follows: Ellipse E_n touches the extremities of the major axis of E_(n-1) and have its focii at the extremities of the minor axis of E_(n-1) . If eccentricity of ellipse E_n is e_n , then the locus of (e_(n)^(2),e_(n-1)^(2)) is

A series of concentric ellipse E_1,E_2,E_3,…,E_n is constructed as follows: Ellipse E_n touches the extremities of the major axis of E_(n-1) and have its focii at the extremities of the minor axis of E_(n-1) If equation of ellipse E_1 is x^(2)/9+y^(2)/16=1 , then equation pf ellipse E_3 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

Consider the ellipse E_1, x^2/a^2+y^2/b^2=1,(a>b) . An ellipse E_2 passes through the extremities of the major axis of E_1 and has its foci at the ends of its minor axis.Consider the following property:Sum of focal distances of any point on an ellipse is equal to its major axis. Equation of E_2 is

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

The difference between the lengths of the major axis and the latus rectum of an ellipse is a. a e b. 2a e c. a e^2 d. 2a e^2

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 .

lim_(n to oo) sum_(r=1)^(n) (1)/(n)e^(r//n) is

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

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