If the tangent at any point of an ellipse `x^2/a^2 + y^2/b^2 =1` makes an angle ` alpha` with the major axis and an angle `beta` with the focal radius of the point of contact then show that the eccentricity ‘e’ of the ellipse is given by the absolute value of `cos beta/cos alpha`

If the tangent at any point of the ellipse (x^(2))/(a^(3))+(y^(2))/(b^(2))=1 makes an angle alpha with the major axis and angle beta with the focal radius of the point of contact,then show that the eccentricity of the ellipse is given by e=cos(beta)/(cos alpha)

If alpha, beta are the eccentric angles of the extermities of a focal chord of an ellipse, then eccentricity of the ellipse is

If the eccentric angles of two points P and Q on the ellipse x^2/a^2+y^2/b^2 are alpha,beta such that alpha +beta=pi/2 , then the locus of the point of intersection of the normals at P and Q is

If alpha and beta are the eccentric angles of the extremities of a focal chord of an ellipse,then prove that the eccentricity of the ellipse is (sin alpha+sin beta)/(sin(alpha+beta))

The tangent at the point alpha on the ellipse x^2/a^2+y^2/b^2=1 meets the auxiliary circle in two points which subtends a right angle at the centre, then the eccentricity 'e' of the ellipse is given by the equation (A) e^2(1+cos^2alpha)=1 (B) e^2(cosec^2alpha-1)=1 (C) e^2(1+sin^2alpha)=1 (D) e^2(1+tan^2alpha)=1

If a tangent to the ellipse x^(2)+4y^(2)=1 makes angle (pi)/(3) with the focal distance of the point of contact then the square of slope of tangent is

Tangents are drawn to the ellipse (x^(2))/(a^(2))+(y^(2))/(b^(2))=1 at two points whose eccentric angles are alpha-beta and alpha+beta The coordinates of their point of intersection are

If the tangents drawn from a point on the hyperbola x^(2)-y^(2)=a^(2)-b^(2) to ellipse (x^(2))/(a^(2))+(y^(2))/(b^(2))=1 make angle alpha and beta with the transverse axis of the hyperbola, then

If two normals drawn from any point to the parabola y^(2) = 4ax make angle alpha and beta with the axis such that tan alpha . tan beta = 2, then find the locus of this point,

The tangent and normal at any point P of an ellipse x^(2)/a^(2)+y^(2)/b^(2)=1 cut its major axis in point Q and R respectively. If QR=a prove that the eccentric angle of the point P is given by e^(2)cos^(2)phi+cosphi-1=0