If the tangents drawn from a point $P$ to the ellipse $4x^2+9y^2-16x+54y+61=0$ are perpendicular,then the locus of $P$ is

The eccentricity of the conic $36x^2 + 144y^2 - 36x - 96y - 119 = 0$ is

The curve represented by $x = 5(\cos t + \sin t)$ and $y = 3(\cos t - \sin t)$ is (where $t$ is a parameter):

The eccentricity of the ellipse $x^2+4 y^2+2 x+16 y+13=0$ is

Let the length of the latus rectum of an ellipse with its major axis along the $x$-axis and center at the origin be $8$. If the distance between the foci of this ellipse is equal to the length of its minor axis,then which one of the following points lies on it?

$A$ tangent having slope of $-\frac{4}{3}$ to the ellipse $\frac{x^2}{18} + \frac{y^2}{32} = 1$ intersects the major and minor axes in points $A$ and $B$ respectively. If $C$ is the centre of the ellipse,then the area of the triangle $ABC$ is: .............. $sq. \,units$