The tangent at a point `P(acosvarphi,bsinvarphi)` of the ellipse `(x^2)/(a^2)+(y^2)/(b^2)=1` meets its auxiliary circle at two points, the chord joining which subtends a right angle at the center. Find the eccentricity of the ellipse.

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

The eqation of auxiliary circle of the ellipse (x^(2))/(a^(2))+(y^(2))/(b^(2)) = 1 is _

If the tangent at any point of the 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 of the ellipse is given by e=cosbeta/(cosalpha)

From any point on any directrix of the ellipse (x^2)/(a^2)+(y^2)/(b^2)=1,a > b , a pari of tangents is drawn to the auxiliary circle. Show that the chord of contact will pass through the correspoinding focus of the ellipse.

Find the locus of the mid point of the chord of the circle x^2+y^2=a^2 which subtend a right angle at the point (0,0).

Find the points on the ellipse (x^2)/(a^2)+(y^2)/(b^2)=1 such that the tangent at each point makes equal angles with the axes.

The locus of the midpoints of chords of the circle x^(2) + y^(2) = 1 which subtends a right angle at the origin is

Find the locus of a point, so that the join of (-5,1) and (3,2) subtends a right angle at the moving point.

Prove that area common to ellipse (x^2)/(a^2)+(y^2)/(b^2)=1 and its auxiliary circle x^2+y^2=a^2 is equal to the area of another ellipse of semi-axis a and a-b.

Find the locus of the middle points of the chords of the parabola y^2=4a x which subtend a right angle at the vertex of the parabola.