Let $P$ be an arbitrary point on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ $a > b > 0$. Suppose $F_1$ and $F_2$ are the foci of the ellipse. The locus of the centroid of the $\Delta P F_1 F_2$ as $P$ moves on the ellipse is

An ellipse has eccentricity $\frac{1}{2}$ and one focus at the point $P\left( {\frac{1}{2},\;1} \right)$. Its one directrix is the common tangent nearer to the point $P$, to the circle ${x^2} + {y^2} = 1$ and the hyperbola ${x^2} – {y^2} = 1$. The equation of the ellipse in the standard form, is

An ellipse is drawn by taking a diameter of the circle ${\left( {x – 1} \right)^2} + {y^2} = 1$ as its semi-minor axis and a diameter of the circle ${x^2} + {\left( {y – 2} \right)^2} = 4$ is semi-major axis. If the center of the ellipse is at the origin and its axes are the coordinate axes, then the equation of the ellipse is :

Find the equation for the ellipse that satisfies the given conditions: Major axis on the $x-$ axis and passes through the points $(4,\,3)$ and $(6,\,2)$

If $\theta $ and $\phi $ are eccentric angles of the ends of a pair of conjugate diameters of the ellipse $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$, then $\theta – \phi $ is equal to