The tangent at point $$(a\cos \theta ,b\sin \theta ),0 < \theta < {\pi \over 2}$$, to the ellipse $${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$$ meets the x-axis at T and y-axis at T$$_1$$. Then the value of $$\mathop {\min }\limits_{0 < \theta < {\pi \over 2}} (OT)(O{T_1})$$ is

Let $$A(2\sec \theta ,3\tan \theta )$$ and $$B(2\sec \phi ,3\tan \phi )$$ where $$\theta + \phi = {\pi \over 2}$$ be two points on the hyperbola $${{{x^2}} \over 4} - {{{y^2}} \over 9} = 1$$. If ($$\alpha,\beta$$) is the point of intersection of normals to the hyperbola at A and B, then $$\beta$$ is equal to

If the lines joining the focii of the ellipse $${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$$ where $$a > b$$, and an extremity of its minor axis is inclined at an angle 60$$^\circ$$, then the eccentricity of the ellipse is

Let $$P(3\sec \theta ,2\tan \theta )$$ and $$Q(3\sec \phi ,2\tan \phi )$$ be two points on $${{{x^2}} \over 9} - {{{y^2}} \over 4} = 1$$ such that $$\theta + \phi = {\pi \over 2},0 < \theta ,\phi < {\pi \over 2}$$. Then the ordinate of the point of intersection of the normals at P and Q is