The equation of the tangent to the circle at the point $(1, -1)$,whose center is the point of intersection of the straight lines $x - y = 1$ and $2x + y = 3$,is:

The angle between the tangents drawn from the origin to the circle $(x-7)^2+(y+1)^2=25$ is

If the tangent at $(1,7)$ to the curve $x^2=y-6$ touches the circle $x^2+y^2+16x+12y+C=0$,then $C=$

If two tangents are drawn from the point $P$ on the circle $x^2+y^2=4$ to the circle $x^2+y^2=1$,where the point $P$ is given by $(\sqrt{2}, \sqrt{2})$,then the slopes of the tangents are:

If the area of the triangle formed by the positive $x-$axis,the normal and the tangent to the circle $(x-2)^{2}+(y-3)^{2}=25$ at the point $(5,7)$ is $A$,then $24A$ is equal to ...... .

Let the lengths of intercepts on the $x$-axis and $y$-axis made by the circle $x^{2}+y^{2}+ax+2ay+c=0$ $(a < 0)$ be $2\sqrt{2}$ and $2\sqrt{5}$,respectively. Then the shortest distance from the origin to a tangent to this circle which is perpendicular to the line $x+2y=0$ is equal to: