Let $S \equiv x^2+y^2-8x+10y+5=0$ be a circle. Let $P(1,1)$ and $Q(1,-1)$ be two points. Then the point of intersection of the polar of $P$ with respect to $S=0$ and the chord with $Q$ as mid-point to $S=0$ is

The inverse of the point $(1, 2)$ with respect to the circle $x^2 + y^2 - 4x - 6y + 9 = 0$ is

If the poles of the line $x-y=0$ with respect to the circles $x^2+y^2-2g_ix+c_i^2=0$ $(i=1, 2, 3)$ are $(\alpha_i, \beta_i)$,then $\sum_{i=1}^3 \frac{\alpha_i+\beta_i}{g_i}=$

The normal to a circle $S=0$ at $P(1,3)$ is $x+2y=7$ and it has another normal at $Q(3,5)$ which is the polar of the point $A(7, -1/2)$ with respect to the circle $x^2+y^2-4x+6y-12=0$. Then,the equation of the circle $S=0$ is

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

Consider the following statements:
$I$. If $P(x_1, y_1)$ and $Q(x_2, y_2)$ are conjugate points with respect to the circle $x^2+y^2+2gx+2fy+c=0$,then $x_1x_2+y_1y_2+g(x_1+x_2)+f(y_1+y_2)+c=0$.
$II$. The pole of the line $x+y+1=0$ with respect to the circle $x^2+y^2=9$ is $(9, 9)$.
Which of the following is true?