If PSQ + PHR be two chords of an ellipse through its foci S and H, then prove that `(PS)/(SQ) + (PH)/(HQ) = 2(1 + e^2)/(1-e^2)`, Where e is the eccentricity of ellipse.

If the normal at one end of lotus rectum of an ellipse passes through one end of minor axis then prove that, e^(2)=(sqrt(5)-1)/(2) , where e is the eccentricity of the ellipse.

If theta and phi are the eccentric angles of the end points of a chord which passes through the focus .of an ellipse x^2/a^2+y^2/b^2=1 .Show that tan(0//2)tan(phi//2)=((e-1)/(e+1)) ,where e is the eccentricity of the ellipse.

If PSQ is focal chord of parabola y^(2)=4ax(a>0), where S is focus then prove that (1)/(PS)+(1)/(SQ)=(1)/(a)

Show that the line (ax)/(3)+(by)/(4)=c be a normal to the ellipse (x^2)/(a^2)+(y^2)/(b^2)=1 , When 15c=a^(2)e^(2) where e is the eccentricity of the ellipse.

If the line (ax)/(3)+(by)/(4)= cbe a normal to the ellipse (x^(2))/(a^(2))+(y^(2))/(a^(2))=1, show that 5c=a^(2)e^(2) where e is the eccentricity of the ellipse.

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 minor axis, then e^(4)+e^(2)= ( where e is the eccentricity of the ellipse)

A circle has same centre as an ellipse and passing through foci S_(1) and S_(2) of the ellipse. The two curves cut in four points. Let P be one of the points. If the Area of /_\PS_(1)S_(2) is 24 and major axis of the ellipse is 14. If the eccentricity of the ellipse is e, then the value of 7e is equal to

S_(1),S_(2), are foci of an ellipse of major axis of length 10 units and P is any point on the ellipse such that perimeter of triangle PS_(1)S_(2) is 15. Then eccentricity of the ellipse is:

Consider an ellipse (x^(2))/(a^(2))+(y^(2))/(b^(2))=1(a gt b) . A hyperbola has its vertices at the extremities of minor axis of the ellipse and the length of major axis of the ellipse is equal to the distance between the foci of hyperbola. Let e_(1) and e_(2) be the eccentricities of ellipse and hyperbola, respectively. Also, let A_(1) be the area of the quadrilateral fored by joining all the foci and A_(2) be the area of the quadrilateral formed by all the directries. The relation between e_(1) and e_(2) is given by