If the normals at `P(theta)` and `Q(pi/2+theta)` to the ellipse `(x^2)/(a^2)+(y^2)/(b^2)=1` meet the major axis at `Ga n dg,` respectively, then `P G^2+Qg^2=` `b^2(1-e^2)(2-e)^2` `a^2(e^4-e^2+2)` `a^2(1+e^2)(2+e^2)` `b^2(1+e^2)(2+e^2)`

If the normal at the end of latus rectum of the ellipse (x^(2))/(a^(2))+(y^(2))/(b^(2))=1 passes through (0,b) then e^(4)+e^(2) (where e is eccentricity) equals

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

Find the area bounded by the ellipse (x^2)/(a^2)+(y^2)/(b^2)=1 and the ordinates x" "=" "0 andx" "=" "a e , where, b^2=a^2(1-e^2) ande" "<" "1 .

The normal at an end of a latus rectum of the ellipse (x^(2))/(a^(2))+(y^(2))/(b^(2))=1 passes through an end of the minor axis if (A)e^(4)+e^(2)=1(B)e^(3)+e^(2)=1(C)e^(2)+e=1(D)e^(3)+e=1

If e_(1)ande_(2) be the eccentricities of the ellipses (x^(2))/(a^(2))+(4y^(2))/(b^(2))=1and(x^(2))/(a^(2))+(4y^(2))/(b^(2))=1 respectively then prove that 3=4e_(2)^(2)-e_(1)^(2) .

If the normal at an end of a latus-rectum of an elipse (x^(2))/(a^(2))+(y^(2))/(b^(2))=1 passes through one extremity of the minor axis,show that the eccentricity of the ellipse is given by e^(4)+e^(-1)=0

If e_1a n d e_2 are respectively the eccentricities of the ellipse (x^2)/(18)+(y^2)/4=1 and the hyperbola (x^2)/9-(y^2)/4=1, then the relation between e_1a n d e_2 is a.2e_1 ^2+e_2 ^2=3 b. e_1 ^2+2e_2^ 2=3 c. 2e_1^ 2+e_2 ^2=3 d. e_1 ^2+3e_2 ^2=2

Normal are drawn to the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 at point theta_(1) and theta_(2) meeting the conjugate axis at G_(1) and G_(2), respectively.If theta_(1)+theta_(2)=(pi)/(2), prove that CG_(1).CG_(2)=(a^(2)e^(4))/(e^(2)-1) where C is the center of the hyperbola and e is the eccentricity.

If e_(1) is the eccentricity of the ellipse (x^(2))/(16)+(y^(2))/(b^(2))=1 and e_(2) is the eccentricity of the hyperbola (x^(2))/(9)-(y^(2))/(b^(2))=1 and e_(1)e_(2)=1 then b^(2) is equal to