The locus of the point of intersection of mutually perpendicular tangent to the ellipse $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$, is

An ellipse with its minor and major axis parallel to the coordinate axes passes through $(0,0),(1,0)$ and $(0,2)$. One of its foci lies on the $Y$-axis. The eccentricity of the ellipse is

If two tangents drawn from a point $(\alpha, \beta)$ lying on the ellipse $25 x^{2}+4 y^{2}=1$ to the parabola $y^{2}=4 x$ are such that the slope of one tangent is four times the other, then the value of $(10 \alpha+5)^{2}+\left(16 \beta^{2}+50\right)^{2}$ equals

Planet $M$ orbits around its sun, $S$, in an elliptical orbit with the sun at one of the foci. When $M$ is closest to $S$, it is $2\,unit$ away. When $M$ is farthest from $S$, it is $18\, unit$ away, then the equation of motion of planet $M$ around its sun $S$, assuming $S$ at the centre of the coordinate plane and the other focus lie on negative $y-$ axis, is

Consider an ellipse with foci at $(5,15)$ and $(21,15)$. If the $X$-axis is a tangent to the ellipse, then the length of its major axis equals

If the normal at one end of the latus rectum of an ellipse  $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$ passes through one end of the minor axis then :