If $A$ and $B$ are the points of intersection of the circle $x^2+y^2-8 x=0$ and the hyperbola $\frac{x^2}{9}-\frac{y^2}{4}=1$ and $a$ point $P$ moves on the line $2 x-3 y+4=0$, then the centroid of $\triangle P A B$ lies on the line :

Find the coordinates of the foci and the vertices, the eccentricity, and the length of the latus rectum of the hyperbola $\frac{y^{2}}{9}-\frac{x^{2}}{27}=1$

The locus of the point of intersection of the lines $(\sqrt{3}) kx + ky -4 \sqrt{3}=0$ and $\sqrt{3} x-y-4(\sqrt{3}) k=0$ is a conic, whose eccentricity is .............

The straight line $x + y = \sqrt 2 p$ will touch the hyperbola $4{x^2} - 9{y^2} = 36$, if

The eccentricity of the hyperbola conjugate to ${x^2} - 3{y^2} = 2x + 8$ is

Point from which two distinct tangents can be drawn on two different branches of the hyperbola $\frac{{{x^2}}}{{25}} - \frac{{{y^2}}}{{16}} = \,1$ but no two different tangent can be drawn to the circle $x^2 + y^2 = 36$ is