If the tangents drawn from a point $P$ to the ellipse $4 x^2+9 y^2-24 x+36 y=0$ are perpendicular,then the locus of $P$ is

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

If the normal at one end of a latus rectum of the ellipse $\frac{x^2}{32}+\frac{y^2}{b^2}=1$ passes through one end of the minor axis,then $\frac{e^4}{1-e^2}=$ (Here $e$ is the eccentricity of the ellipse)

The locus of the point of intersection of two perpendicular tangents to the ellipse $\frac{x^2}{9} + \frac{y^2}{4} = 1$ is:

The tangent to the ellipse $9x^2 + 16y^2 = 288$ making equal intercepts on the coordinate axes intersects the $X$-axis and the $Y$-axis at points $A$ and $B$ respectively. Then,the area of $\triangle OAB$ (where $O$ is the origin) is:

The ratio of the distance of a point from a fixed point and a line $x = 9/2$ is always $2 : 3$. Then the locus of the point is