Tangents are drawn from the point $(-1,-4)$ to the circle $x^2 + y^2 - 2x + 4y + 1 = 0$. Length of corresponding chord of contact will be-

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

If variable point $(x, y)$ satisfies the equation $x^2 + y^2 -8x -6y + 9 = 0$ , then range of $\frac{y}{x}$ is

Let $O$ be the centre of the circle $x ^2+ y ^2= r ^2$, where $r >\frac{\sqrt{5}}{2}$. Suppose $P Q$ is a chord of this circle and the equation of the line passing through $P$ and $Q$ is $2 x+4 y=5$. If the centre of the circumcircle of the triangle $O P Q$ lies on the line $x+2 y=4$, then the value of $r$ is. . . .

If $a > 2b > 0$ then the positive value of m for which $y = mx - b\sqrt {1 + {m^2}} $ is a common tangent to ${x^2} + {y^2} = {b^2}$ and ${(x - a)^2} + {y^2} = {b^2}$, is

The equation of the normal at the point $(4,-1)$ of the circle $x^2+y^2-40 x+10 y=153$ is