The equations of the tangents drawn from the point $(0, 1)$ to the circle $x^2 + y^2 - 2x + 4y = 0$ are:

$A$ circle with centre $(a, b)$ passes through the origin. The equation of the tangent to the circle at the origin is

Two sides of a square are along the lines $x=-5$ and $y=4$. The point of intersection of the diagonals is $(3,-4)$. The point of intersection of the tangents drawn to the circumcircle of the square at the two consecutive vertices lying on $x=-5$ is

The product of the slopes of the common tangents drawn to the circles $x^2+y^2+2x-2y-2=0$ and $x^2+y^2-2x+2y+1=0$ is:

If $m_1, m_2$ are the slopes of the tangents drawn from a point $(1, -3)$ to the circle $x^2+y^2-6x+4y+12=0$,then $9(m_1^2+m_2^2) = $

$A$ line is a common tangent to the circle $(x-3)^{2}+y^{2}=9$ and the parabola $y^{2}=4x$. If the two points of contact $(a, b)$ and $(c, d)$ are distinct and lie in the first quadrant,then $2(a+c)$ is equal to ........ .