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:

If the line $4x - 3y + 7 = 0$ touches the circle $x^2 + y^2 - 6x + 4y - 12 = 0$ at $(\alpha, \beta)$,then $\alpha + 2\beta =$

If the straight line $y = mx + c$ touches the circle ${x^2} + {y^2} - 4y = 0$,then the value of $c$ will be

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 $O$ be the origin and $OP$ and $OQ$ be the tangents to the circle $x^2+y^2-6x+4y+8=0$ at the points $P$ and $Q$ on it. If the circumcircle of the triangle $OPQ$ passes through the point $(\alpha, \frac{1}{2})$,then a value of $\alpha$ is

The length of the tangent from the point $(4, 5)$ to the circle $x^2 + y^2 + 2x - 6y - 6 = 0$ is