A variable point P on the ellipse of eccentricity e is joined to the foci `S` and `S'`. The eccentricity of the locus of incentre of the triangle `PSS'` is (A) `sqrt((2e)/(1+e))` (B) `sqrt(e/(1+e))` (C) `sqrt((1-e)/(1+e))` (D) `e/(2(1+e))`

The normal at a variable point P on the ellipse (x^2)/(a^2)+(y^2)/(b^2)=1 of eccentricity e meets the axes of the ellipse at Qa n dRdot Then the locus of the midpoint of Q R is a conic with eccentricity e ' such that (a) e^(prime) is independent of e (b) e^(prime)=1 (c) e^(prime)=e (d) e^(prime)=1/e

Two poles are a metres apart and the height of one is double of the other. If from the middle point of the line joining their feet an observer finds the angular elevations of their tops to be complementary, then the height of the smaller is (a) sqrt(2)am e t r e s (b) a/(2sqrt(2))m e t r e s (c) a/(sqrt(2))m e t r e s (d) 2am e t r e s

If e and e' be the eccentricities of two conics S=0 and S'=0 and if e^(2)+e'^(2)=3 , then both S and S' can be

The sum of the series 1+1/4.2!1/16.4!+1/64.6!+………to oo is (A) (e+1)/(2sqrt(e)) (B) (e-1)/sqrt(e) (C) (e-1)/(2sqrt(e)) (D) (e+1)/2sqrt(e)

The normal at P to a hyperbola of eccentricity e , intersects its transverse and conjugate axes at L and M respectively. Show that the locus of the middle point of LM is a hyperbola of eccentricity e/sqrt(e^2-1)

S and T are the foci of the ellipse x^2/a^2+y^2/b^2 = 1 and B is an end of the minor axis. If STB is an equilateral triangle, the eccentricity of the ellipse is e then find value of 4e

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 sum of the series 1+1/4.2!1/16.4!+1/64.6!+………to oo is (A) (e+1)/2sqrt(e) (B) (e-1)/sqrt(e) (C) (e-1)/+2sqrt(e) (D) (e-1)/sqrt(e)

Let Sa n dS ' be two foci of the ellipse (x^2)/(a^2)+(y^2)/(b^2)=1 . If a circle described on S S^(prime) as diameter intersects the ellipse at real and distinct points, then the eccentricity e of the ellipse satisfies (a) e=1/(sqrt(2)) (b) e in (1/(sqrt(2)),1) (c) e in (0,1/(sqrt(2))) (d) none of these

If alpha and beta are eccentric angles of the ends of a focal chord of the ellipse x^2/a^2 + y^2/b^2 =1 , then tan alpha/2 .tan beta/2 is (A) (1-e)/(1+e) (B) (e+1)/(e-1) (C) (e-1)/(e+1) (D) none of these