A circle has the same center as an ellipse and passes through the foci `F_1a n dF_2` of the ellipse, such that the two curves intersect at four points. Let `P` be any one of their point of intersection. If the major axis of the ellipse is 17 and the area of triangle `P F_1F_2` is 30, then the distance between the foci is

If the minor axis of an ellipse is equal to the distance between its foci, prove that its eccentricity is 1/sqrt(2) .

If the tangent to the ellipse x^2+2y^2=1 at point P(1/(sqrt(2)),1/2) meets the auxiliary circle at point R and Q , then find the points of intersection of tangents to the circle at Q and Rdot

P is a variable point on the ellipse with foci S_1 and S_2 . If A is the area of the the triangle PS_1S_2 , the maximum value of A is

Find the equation of the ellipse that passes through the origin and has the foci at the points (-1, 1) and S\'(1,1) .

Let P be a variable point on the ellipse (x^2)/(a^2)+(y^2)/(b^2)=1 with foci F_1" and "F_2 . If A is the area of the trianglePF_1F_2 , then the maximum value of A is

Let P be a variable point on the ellipse (x^2)/(a^2)+(y^2)/(b^2)=1 with foci F_1" and "F_2 . If A is the area of the trianglePF_1F_2 , then the maximum value of A is

Let P be a variable point on the ellipse with foci S_(1) and S_(2) . If A be the area of trianglePS_(1)S_(2) then find the maximum value of A

The radius of the circle passing through the foci of the ellipse (x^(2))/(16) + (y^(2))/( 9) = 1 and having its centre at (0,3) is

Suppose an ellipse and a hyperbola have the same pair of foci on the x -axis with centres at the origin and they intersect at (2,2) . If the eccentricity of the ellipse is (1)/(2) , then the eccentricity of the hyperbola, is

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