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`

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

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

If e_1 is eccentricity of the ellipse x^2/16+y^2/7=1 and e_2 is eccentricity of the hyperbola x^2/9-y^2/7=1 , then e_1+e_2 is equal to:

If the length of latusrectum of ellipse E_(1):4(x+y+1)^(2)+2(x-y+3)^(2)=8 and E_(2)=(x^2)/(p)+(y^2)/(p^2)=1, (0ltplt1) are equal , then area of ellipse E_(2) , is

If a,b,c,d ,e are +ve real numbers such that a+b+c+d+e=8 and a^2 + b^2 +c^2 + d^2 +e^2 = 16 , then the range of 'e' is

The slope of the tangent to the curve y^2e^(xy) =9e^(-3)x^2 at (-1,3) is

Two concentric ellipses are such that the foci of one are on the other and their major axes are equal. Let ea n de ' be their eccentricities. Then. the quadrilateral formed by joining the foci of the two ellipses is a parallelogram the angle theta between their axes is given by theta=cos^(-1)sqrt(1/(e^2)+1/(e^('2))=1/(e^2e^('2))) If e^2+e^('2)=1, then the angle between the axes of the two ellipses is 90^0 none of these

If radii of director circle of the ellipse (x^(2))/(a^(2))+(y^(2))/(b^(2))=1 and hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 are in the ratio 1:3 and 4e_(1)^(2)-e_(2)^(2)=lambda , where e_1 and e_2 are the eccetricities of ellipse and hyperbola respectively, then the value of lambda is

The graph of the conic x^(2)-(y-1)^(2)=1 has one tangent line with positive slope that passes through the origin. The point of tangency being (a, b). Q. If e be the eccentricity of the conic, then the value of (1+e^(2)+e^(4)) is