Let the circle with centre at origin pass through the vertices of an equilateral triangle $ABC$. If $A = (2, 4)$,then the length of the median through $A$ is

Find the equation of the chord of the circle $x^2 + y^2 - 6x + 8y = 0$ which is bisected at the point $(5, -3)$.

Statement $(A):$ The number of common tangents to the circles $x^2 + y^2 = 4$ and $x^2 + y^2 - 6x - 8y = 24$ is $4$.
Reason $(R):$ For two circles with centers $C_1, C_2$ and radii $r_1, r_2$,if $|C_1C_2| > r_1 + r_2$,then the circles have $4$ common tangents.

If the angle between the circles $x^2+y^2-12x-6y+41=0$ and $x^2+y^2+kx+6y-59=0$ is $45^{\circ}$,then a value of $k$ is

If two vertices of an equilateral triangle are $(-1, 0)$ and $(1, 0)$,then its circumcircle is:

Let the tangents drawn from $P(-1, -1)$ to the circle $x^2 + y^2 - 2x - 4y - 4 = 0$ touch the circle at the points $A$ and $B$. Then the area of the triangle $PAB$ (in square units) is