Chords of the ellipse `(x^2)/(a^2)+(y^2)/(b^2)=1` are drawn through the positive end of the minor axis. Then prove that their midpoints lie on the ellipse.

If the normal at one end of the latus rectum of the ellipse (x^2)/(a^2)+(y^2)/(b^2)=1 passes through one end of the minor axis, then prove that eccentricity is constant.

Find the equation of the normal to the ellipse (x^2)/(a^2)+(y^2)/(b^2)=1 at the positive end of the latus rectum.

Find the locus of the middle points of chord of an ellipse x^2/a^2 + y^2/b^2 = 1 which are drawn through the positive end of the minor axis.

P and Q 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 P B Q is an equilateral triangle, then the eccentricity of the ellipse is

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 . . .

If two points are taken on the minor axis of an ellipse (x^2)/(a^2)+(y^2)/(b^2)=1 at the same distance from the center as the foci, then prove that the sum of the squares of the perpendicular distances from these points on any tangent to the ellipse is 2a^2dot

A parabola is drawn with focus at one of the foci of the ellipse (x^2)/(a^2)+(y^2)/(b^2)=1 . If the latus rectum of the ellipse and that of the parabola are same, then the eccentricity of the ellipse is (a) 1-1/(sqrt(2)) (b) 2sqrt(2)-2 (c) sqrt(2)-1 (d) none of these

if the line x- 2y = 12 is tangent to the ellipse (x^(2))/(a^(2))+(y^(2))/(b^(2))=1 at the point (3,(-9)/(2)) then the length of the latusrectum of the ellipse is

The value of a for the ellipse (x^2)/(a^2)+(y^2)/(b^2)=1,(a > b), if the extremities of the latus rectum of the ellipse having positive ordinates lie on the parabola x^2=2(y-2) is ___

A O B is the positive quadrant of the ellipse (x^2)/(a^2)+(y^2)/(b^2)=1 in which O A=a ,O B=b . Then find the area between the arc A B and the chord A B of the ellipse.