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: ................ $^o$

Find the equation for the ellipse that satisfies the given conditions: $b=3,\,\, c=4,$ centre at the origin; foci on the $x$ axis.

In an ellipse, its foci and ends of its major axis are equally spaced. If the length of its semi-minor axis is $2 \sqrt{2}$, then the length of its semi-major axis is

Let $P(a\sec \theta ,\;b\tan \theta )$ and $Q(a\sec \varphi ,\;b\tan \varphi )$, where $\theta + \phi = \frac{\pi }{2}$, be two points on the hyperbola $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1$. If $(h, k)$ is the point of intersection of the normals at $P$ and $Q$, then $k$ is equal to

The equation of the ellipse whose one focus is at $(4, 0)$ and whose eccentricity is $4/5$, is

The co-ordinates of the foci of the ellipse $3{x^2} + 4{y^2} - 12x - 8y + 4 = 0$ are