Show that the point of intersection of the tangents to the ellipse `x^2/a^2+y^2/b^2=1( a gt b)` which are inclined at an angle `theta_1 and theta_2` with its major axis such that `cot theta_1+ cot theta_2=k^2` lies on the curve `k^2(y^2-b^2)=2xy`

Show that the point o f intersection of the tangents to the ellipse (x^(2))/(a^(2))+(y^(2))/(b^(2)) = l( a gt b) which are inclined at an angle theta , and theta_(2) with its major axis such that cot theta_(1), + cot theta_(2), = k^(2) lies on the curve k^(2) (y^(2) ~-b^(2))= 2xy .

Find the locus of point of intersection of tangents to the ellipse x^2/a^2+y^2/b^2=1 which are inclined at an angle alpha with each other.

The locus of the point of intersection of two tangents to the parabola y^(2)=4ax which make the angles theta_(1) and theta_(2) with the axis so that cot theta_(1)+cot theta_(2) = k is

The locus of the point of intersection of two tangents to the parabola y^(2)=4ax which make the angles theta_(1) and theta_(2) with the axis so that tan theta_(1) tan theta_(2) =k is

The length of the subtangent (if exists) at any point theta on the ellipse x^2/a^2+y^2/b^2=1 is