Pole and Polar

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

If the polar of a point $P$ with respect to a circle of radius $r$ which touches the coordinate axes and lies in the first quadrant is $x+2y=4r$, then the point $P$ is

The polars of two points $A(5,7)$ and $B(3,3)$ with respect to the circle $x^2+y^2+2gx+2fy+c=0$ intersect at $C$ then find the polar of $C$ with respect to the circle.

The polars of two points $A(1,3)$ and $B(2,1)$ with respect to the circle $x^2+y^2+2gx+2fy+c=0$ intersect at $C$ then find the polar of $C$ with respect to the circle.

The equation of polar of $(1,1)$ with respect to the circle $x^2+y^2+4x+6y-3=0$ is

The polars of a point $\mathrm{P}$ with respect to two fixed circles meet in the point $\mathrm{Q}$. Prove that the circle on $\mathrm{P}\mathrm{Q}$ as diameter passes through two fixed points, and cuts both the given circles at right angles.

If the lines $kx+2y-4=0$ and $5x-2y-4=0$ are conjugate with respect to the circle $x^2+y^2-2x-2y+1=0$ , then $k=$

If $\left(4,2\right)$ and $\left(k,-3\right)$ are conjugate points with respect to $x^2+y^2-5x+8y+6=0,$ then $k=$

The condition for the lines $lx+my+n=0$ and $l_{1}x+m_{1}y+n_1=0$ to be conjugate with respect to the circle $x^2+y^2=r^2,$ is

If $\left(1, a\right), (b, 2)$ are conjugate points with respect to the circle $x^2+y^2=25$, then $4a+2b$ is equal to

The Polar equation of the circle with centre $\left(2,\frac{\pi }{2}\right)$ and radius $3$ units is