A series of concentric ellipse E_1,E_2,E_3,…,E_n is constructed as follows: Ellipse E_n touches...

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 (A) sqrt 5/3 (B) (sqrt 5-1)/2 (C) (sqrt 5 +1)/2 (D) 1/sqrt5

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_(1) and e_(2) are the eccentricities of the ellipse and the hyperbola,respectively, then prove that (1)/(e_(1)^(2))+(1)/(e_(2)^(2))=2

Define the collection {E_1,E_2,E_3,"....."} of ellipses and {R_1,R_2,R_3,"....."} of rectangles as follows: E_1= (x^2)/(9)+(y^2)/(4)=1 , R_1 : rectangle of largest area, with sides parallel to the axes, inscribed in E_1 , E_n : ellipse (x^2)/(a_(n)^(2))+(y^2)/(b_(n)^(2))=1 of largest are inscribed in R_(n-1), n gt 1 . then which of the following options is/are corrct?

The eccentricity e of an ellipse satisfies the condition :

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

The eccentricity of the ellipse 9x^2+25y^2=225 is e then the value of 5e is