Let ` E_1 and E_2` be two ellipse whose centers are at the origin. The major axes of `E_1 and E_2` 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,`E_1 and E_2` at P,Q and R respectively. Suppose that `PQ=PR=(2sqrt2)/3`. If `e_1 and e_2` are the eccentricities of `E_1 and E_2`,respectively, then the correct expression (s) is (are)

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):

Let E_(1) and E_(2) be two ellipse whsoe centers are at the origin. The major axes of E_(1) and E_(2) lie along the x-axis , and the y-axis, respectively. Let S be the circle x^(2)+(y-1)^(2)=2 . The straigth line x+y=3 touches the curves, S, E_(1) and E_(2) at P,Q and R, respectively . Suppose that PQ=PR=(2sqrt(2))/(3) . If e_(1) and e_(2) are the eccentricities of E_(1) and E_(2) respectively, thent hecorrect expression (s) is (are)

If radii of director circles of x^2/a^2+y^2/b^2=1 and x^2/a^2-y^2/b^2=1 are 2r and r respectively, let e_E and e_H are the eccentricities of ellipse and hyperbola respectively, then

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

Let E_1 and E_2 are two mutually exclusive and exhaustive events. If odds are 2:3 against E_1 then odds in favour of E_ 2 are

For the batteries shown in fig. R_1, R_2 and R_3 are the internal resistance of E_1, E_2, and E_3, respectively Then, which of the following is / are correct?

If e_(1) and e_(2) are eccentricities of the hyperbolas xy=c^(2) and x^2-y^(2)=a^(2) then e_(1)^(2)+e_(2)^(2)=

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

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 .

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 .