The line $ax + by + c = 0$ is normal to the circle $x^2 + y^2 + 2gx + 2fy + d = 0$ if

If $y=\sqrt{3}x+k_1$ and $y=\sqrt{3}x+k_2$ are two parallel tangents of a circle of radius $2 \text{ units}$,then $|k_1-k_2|$ is equal to

The equations of the tangents drawn from the origin to the circle $x^2+y^2+2gx+2fy+g^2=0$ are

If the tangents drawn at the points $O(0,0)$ and $P(1+\sqrt{5}, 2)$ on the circle $x^{2}+y^{2}-2x-4y=0$ intersect at the point $Q$,then the area of the triangle $OPQ$ is equal to

If the angle between the pair of tangents drawn to the circle $x^2+y^2-2x+4y+3=0$ from the point $(6,-5)$ is $\theta$,then $\cot \theta=$

If the tangent at the point $P$ on the circle $x^2+y^2+6x+6y=2$ meets the straight line $5x-2y+6=0$ at a point $Q$ on the $Y$-axis,then the length of $PQ$ is