If the points $(2,3)$ and $(K,-2)$ are conjugate with respect to the circle $x^2+y^2-2x+4y-2=0$,then $K=$

The lines $x \cos \alpha + y \sin \alpha = P, \alpha \in R$ are chords of the hyperbola $\frac{x^2}{9} - \frac{y^2}{36} = 1$ and they subtend a right angle at the centre of the hyperbola. The locus of the poles of these lines with respect to the given hyperbola is

If the inverse point of $(1,1)$ with respect to the circle $x^2+y^2-4x-6y+12=0$ is $(h, k)$,then $h+k$ is equal to

For all real values of $k$,the point which lies on the polar of $(k, k+1)$ with respect to the circle $x^2+y^2+4x-8y-5=0$ is

The distance between the polar of $P(2,3)$ with respect to the circle $x^2+y^2-2x-2y+1=0$ and the polar of the inverse point of $P$ with respect to the same circle is

The pole of the line $\frac{x}{a} + \frac{y}{b} = 1$ with respect to the circle $x^2 + y^2 = c^2$ is