If $\frac{\sqrt{3}}{a}x + \frac{1}{b}y = 2$ touches the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$,then its eccentric angle $\theta$ is equal to: ................ $^o$

The longest distance of the point $(a, 0)$ from the curve $2x^2+y^2=2x$ is

The equations of the tangents to the ellipse $\frac{x^2}{16} + \frac{y^2}{9} = 1$ making an inclination of $30^\circ$ with the major axis are

The tangent at point $(a \cos \theta, b \sin \theta)$, where $0 < \theta < \frac{\pi}{2}$, to the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ meets the $x$-axis at $T$ and the $y$-axis at $T_1$. Then the value of $\min_{0 < \theta < \frac{\pi}{2}} (OT)(OT_1)$ is

$A$ chord $PQ$ of the ellipse $\frac{x^2}{9} + \frac{y^2}{4} = 1$ subtends a right angle at its centre. The locus of the point of intersection of tangents drawn at $P$ and $Q$ is-

The lengths of the major and minor axes of an ellipse are $10$ and $8$ respectively,and its major axis lies along the $y$-axis. The equation of the ellipse,with its center at the origin,is: