The equation of an ellipse whose focus $(-1, 1)$, whose directrix is $x - y + 3 = 0$ and whose eccentricity is $\frac{1}{2}$, is given by
$7{x^2} + 2xy + 7{y^2} + 10x - 10y + 7 = 0$
$7{x^2} - 2xy + 7{y^2} - 10x + 10y + 7 = 0$
$7{x^2} - 2xy + 7{y^2} - 10x - 10y - 7 = 0$
$7{x^2} - 2xy + 7{y^2} + 10x + 10y - 7 = 0$
The eccentricity of the ellipse $9{x^2} + 5{y^2} - 30y = 0$, is
If $P$ lies in the first quadrant on the ellipse $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$ (where $a > b$ ), and tangent & normal drawn at $P$ meets major axis at the points $T$ & $N$ respectively, then the value of $\frac{{\left( {\left| {{F_2}N} \right| + \left| {{F_1}N} \right|} \right)\left( {\left| {{F_2}T} \right| - \left| {{F_1}T} \right|} \right)}}{{\left( {\left| {{F_2}N} \right| - \left| {{F_1}N} \right|} \right)\left( {\left| {{F_2}T} \right| + \left| {{F_1}T} \right|} \right)}}$ is equal to (where $F_1$ & $F_2$ are the foci $(ae, 0)$ & $(-ae, 0)$ respectively)
If the distance between the foci of an ellipse be equal to its minor axis, then its eccentricity is
Minimum distance between two points $P$ and $Q$ on the ellipse $\frac{{{x^2}}}{{25}} + \frac{{{y^2}}}{4} = 1$ , if difference between eccentric angles of $P$ and $Q$ is $\frac{{3\pi }}{2}$ , is
A rod of length $12 \,cm$ moves with its ends always touching the coordinate axes. Determine the equation of the locus of a point $P$ on the rod, which is $3\, cm$ from the end in contact with the $x-$ axis.