When will the line $y = x + c$ intersect the circle $x^2 + y^2 = 1$ at two coincident points?

If $3x + y + k = 0$ is a tangent to the circle $x^{2} + y^{2} = 10$,the values of $k$ are

Two tangents are drawn from a point $P$ to the circle $x^{2}+y^{2}-2x-4y+4=0$,such that the angle between these tangents is $\tan^{-1}\left(\frac{12}{5}\right)$,where $\tan^{-1}\left(\frac{12}{5}\right) \in (0, \pi)$. If the centre of the circle is denoted by $C$ and these tangents touch the circle at points $A$ and $B$,then the ratio of the areas of $\Delta PAB$ and $\Delta CAB$ is:

If the line $3x - 4y = 1$ touches the circle $(x - 1)^2 + (y + 2)^2 = 4$ at $(\alpha, \beta)$,the values of $\alpha$ and $\beta$ are

Two tangents to the circle $x^2+y^2=4$ at the points $A$ and $B$ meet at the point $P(-4,0)$. Then the area of the quadrilateral $PAOB$,where $O$ is 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