If the equation of the polar of the point $(\alpha, -1)$ with respect to the circle $x^2+y^2-4x-6y-12=0$ is $y=\beta$,then $4(\alpha+\beta)=$

An ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ with eccentricity $e=\frac{2 \sqrt{2}}{3}$ is inscribed in a circle $x^2+y^2=18$ such that the length of its major axis is equal to the diameter of this circle. The locus of the poles of all the tangents of the circle with respect to the ellipse is

The condition for the lines $lx + my + n = 0$ and $l_1x + m_1y + n_1 = 0$ to be conjugate with respect to the circle $x^2 + y^2 = r^2$ is:

The pole of the line $x-5y-7=0$ with respect to the circle $S \equiv x^2+y^2-2x+4y+1=0$ is $P(a, b)$. If $C$ is the centre of the circle $S=0$,then $PC=$

If $5x + 6y - 34 = 0$ and $2x + y + c = 0$ are conjugate lines with respect to the circle $x^2 + y^2 - 8x - 10y + 25 = 0$,then which of the following points lies on the line $2x + y + c = 0$?

$A$ rectangle is formed by the lines $x=4, x=-2, y=5$,and $y=-2$. $A$ circle is drawn passing through the vertices of this rectangle. The pole of the line $y+2=0$ with respect to this circle is: