show that the length of the focal chord of the ellipse `(x^(2))/(a^(2))+(y^(2))/(b^(2))=1` which makes an angle`theta` with the major axis is `(2ab^(2))/(a^(2) sin ^(2) theta+ b^(2) cos^(2) theta)` unit.

The slop of the tangent to the ellipse (x^(2))/(a^(2))+(y^(2))/(b^(2)) =1 at the point (a cos theta, b sin theta) - is

If x=a sin theta and y=b tan theta , then prove that (a^(2))/(x^(2))-(b^(2))/(y^(2))=1 .

sin2theta - cos2theta =1

If e is the eccentricity of the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 and theta is the angle between the asymptotes, then cos.(theta)/(2) is equal to

int (sin^(2)theta d theta)/((1 + cos theta)^(2))

If omega is one of the angles between the normals to the ellipse (x^2)/(a^2)+(y^2)/(b^2)=1 (b>a) at the point whose eccentric angles are theta and pi/2+theta , then prove that (2cotomega)/(sin2theta)=(e^2)/(sqrt(1-e^2))

The focal chord of the parabola y^(2)=4ax makes an angle theta with its aixs. Show that the length of the chord will be 4acosec^(2)theta .

(1-cos2theta)/(sin2theta)=

If the eccentric angles of the two points on the ellipse (x^(2))/(a^(2)) =(y^(2))/(b^(2)) = 1 (a^(2)gt b^(2)) and theta_(1) and theta_(2) then prove that the equation of chord passing through these two point is (x)/(a)"cos"(theta_(1)+theta_(2))/(2)+(y)/(b)"sin"(theta_(1)+theta_(2))/(2)="cos" (theta_(1)-theta_(2))/(2) .

The tangent at the point theta on the ellipse (x^(2))/(a^(2))+(y^(2))/(b^(2))=1 , meets its auxiliary circle at two points whose join subtends a right angle at the centre, show that the eccentricity of the ellipse is given by, (1)/(e^(2))=1+sin^(2)theta