The maximum area of a rectangle inscribed in the circle $(x+1)^{2}+(y-3)^{2}=64$ is

The length of the diameter of the circle which touches the $x-$axis at the point $(1,0)$ and passes through the point $(2,3)$ is:

The equation of the circle which touches the circle $x^2+y^2-6x+6y+17=0$ externally and has the lines $x^2-3xy-3x+9y=0$ as two normals,is

If the lengths of the tangents drawn from the point $(1,2)$ to the circles $x^2+y^2+x+y-4=0$ and $3x^2+3y^2-x-y-\lambda=0$ are in the ratio $3:4$,then $\lambda$ is equal to

If $(x_i, y_i)$ are the vertices of an equilateral triangle $ABC$ such that $(x_1 - 2)^2 + (y_1 - 3)^2 = (x_2 - 2)^2 + (y_2 - 3)^2 = (x_3 - 2)^2 + (y_3 - 3)^2$,then find the value of $2(x_1 + x_2 + x_3) + 3(y_1 + y_2 + y_3)$.

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