If $A(4,0)$ and $B(-4,0)$ are two points,then the locus of a point $P$ such that $PA - PB = 4$ is

Consider a hyperbola $H$ having its centre at the origin and foci on the $x$-axis. Let $C_1$ be a circle touching the hyperbola $H$ and having its centre at the origin. Let $C_2$ be a circle touching the hyperbola $H$ at its vertex and having its centre at one of its foci. If the areas (in sq. units) of $C_1$ and $C_2$ are $36 \pi$ and $4 \pi$,respectively,then the length (in units) of the latus rectum of $H$ is

If the circle $x^2+y^2=a^2$ intersects the hyperbola $xy=c^2$ in four points $(x_i, y_i)$,for $i=1, 2, 3, 4$,then $y_1+y_2+y_3+y_4$ equals

If $P(x_1, y_1), Q(x_2, y_2), R(x_3, y_3)$ and $S(x_4, y_4)$ are $4$ concyclic points on the rectangular hyperbola $xy = c^2$,the coordinates of the orthocentre of the triangle $PQR$ are:

If the tangent and normal to a rectangular hyperbola $xy = c^2$ at a variable point cut off intercepts $a_1, a_2$ on the $x$-axis and $b_1, b_2$ on the $y$-axis,then $(a_1a_2 + b_1b_2)$ is

The equation of the normal to the hyperbola $\frac{x^2}{16} - \frac{y^2}{9} = 1$ at the point $(8, 3\sqrt{3})$ is