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

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 $\pi$)

Find the minimum and maximum distance from the point $(6, 8)$ to the circle $x^2 + y^2 = 4$.

The line $y=mx+c$ intercepts the circle $x^2+y^2=r^2$ in two distinct points,if

$A$ square is inscribed in the circle $x^2 + y^2 - 2x + 4y + 3 = 0$,whose sides are parallel to the coordinate axes. One vertex of the square is

Let $AB$ be a line segment with mid-point $C$ and $D$ be the mid-point of $AC$. Let $C_1$ be the circle with diameter $AB$ and $C_2$ be the circle with diameter $AC$. Let $E$ be a point on $C_1$ such that $EC$ is perpendicular to $AB$. Let $F$ be a point on $C_2$ such that $DF$ is perpendicular to $AB$ and $E$ and $F$ lie on opposite sides of $AB$. Then,the value of $\sin \angle FEC$ is