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

The length of the chord intercepted by the circle $x^2+y^2+2x+4y-20=0$ on the line $3x+4y-6=0$ is

Let $S=0$ be the circle passing through the points $(2,0)$,$(1,-2)$,and $(-1,1)$. Then the point $(1,2)$

The number of common tangents to the circles ${x^2} + {y^2} - x = 0$ and ${x^2} + {y^2} + x = 0$ is:

The line $2x - 3y = 1$ divides the circular region $x^2 + y^2 \leq 6$ into two parts. If $S = \left\{ \left(2, \frac{3}{4}\right), \left(\frac{5}{2}, \frac{3}{4}\right), \left(\frac{1}{4}, -\frac{1}{4}\right), \left(\frac{1}{8}, \frac{1}{4}\right) \right\}$,then the number of points in the set $S$ that lie inside the smaller part is:

If a circle with centre at $(-1, 1)$ touches the line $x + 2y + 4 = 0$,then the coordinates of the point of contact are: