The chord of the circle $x^{2}+y^{2}-4x=0$ which is bisected at $(1,0)$ is perpendicular to the line

The circumference of a circle passing through the point $(4,6)$ with two normals represented by $2x - 3y + 4 = 0$ and $x + y - 3 = 0$ is (in $\pi$)

The centres of two circles $C_1$ and $C_2$ each of unit radius are at a distance of $6$ units from each other. Let $P$ be the midpoint of the line segment joining the centres of $C_1$ and $C_2$ and $C$ be a circle touching circles $C_1$ and $C_2$ externally. If a common tangent to $C_1$ and $C$ passing through $P$ is also a common tangent to $C_2$ and $C$,then the radius of the circle $C$ is

$x^2+y^2+2x-6y-6=0$ and $x^2+y^2-6x-2y+k=0$ are two intersecting circles and $k$ is not an integer. If $\theta$ is the angle between the two circles and $\cos \theta = \frac{-5}{24}$,then $k=$

If the chord $y = mx + 1$ of the circle ${x^2} + {y^2} = 1$ subtends an angle of measure ${45^\circ}$ at the major segment of the circle,then the value of $m$ is:

The equation of the circumcircle of the triangle formed by the lines $y + \sqrt{3}x = 6$,$y - \sqrt{3}x = 6$,and $y = 0$ is: