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):

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

PARAGRAPH "X" Let S be the circle in the x y -plane defined by the equation x^2+y^2=4. (For Ques. No 15 and 16) Let E_1E_2 and F_1F_2 be the chords of S passing through the point P_0(1,\ 1) and parallel to the x-axis and the y-axis, respectively. Let G_1G_2 be the chord of S passing through P_0 and having slope -1 . Let the tangents to S at E_1 and E_2 meet at E_3 , the tangents to S at F_1 and F_2 meet at F_3 , and the tangents to S at G_1 and G_2 meet at G_3 . Then, the points E_3,\ F_3 and G_3 lie on the curve x+y=4 (b) (x-4)^2+(y-4)^2=16 (c) (x-4)(y-4)=4 (d) x y=4

Two cells of emf E_1 and E_2 are joined in opposition (such that E_1>E_2 ). If r_1 and r_2 be the internal resistances and R be the external resistance, then the terminal potential difference is:

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 .

Find the area bounded by the ellipse (x^(2))/(a^(2)) + (y^(2))/(b^(2)) = 1 and the lines x = 0 and x = ae where b^(2) = a^(2)(1 - e^(2)) and e lt 1 .

Let E be the ellipse (x^2)/9+(y^2)/4=1 and C be the circle x^2+y^2=9 . Let Pa n dQ be the points (1, 2) and (2, 1), respectively. Then (a) Q lies inside C but outside E (b) Q lies outside both Ca n dE (c) P lies inside both C and E (d) P lies inside C but outside E

Find the area enclosed between the curves: y = log_e (x + e) , x = log_e (1/y) & the x-axis.

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

Let S and S' be the fociof the ellipse (x^(2))/(a^(2))+(y^(2))/(b^(2))=1 whose eccentricity is e. P is a variable point on the ellipse. Consider the locus the incenter of DeltaPSS' The eccentricity of the locus of the P is