The line $y=mx+c$ intersects the circle $x^2+y^2=r^2$ at two distinct points if:

The area (in sq. units) of the region bounded by the circle $x^2+y^2=64$, positive $x$-axis and the line $y=\sqrt{3}x$ is

If the circle $x^2+y^2+2gx+2fy+c=0$ $(c>0)$ touches both the coordinate axes and lies in the third quadrant,then the length of the chord intercepted by the circle on the line $x+y+\sqrt{c}=0$ is

The centre of the circle passing through the point $(0, 1)$ and touching the curve $y = x^2$ at $(2, 4)$ is

Two circles with equal radii intersect at the points $(0, 1)$ and $(0, -1).$ The tangent at the point $(0, 1)$ to one of the circles passes through the centre of the other circle. Then the distance between the centres of these circles is

Let the equation of the circle,which touches the $x$-axis at the point $(a, 0), a > 0$ and cuts off an intercept of length $b$ on the $y$-axis,be $x^2 + y^2 - \alpha x + \beta y + \gamma = 0$. If the circle lies below the $x$-axis,then the ordered pair $(2a, b^2)$ is equal to: