If $(a \cos \theta_i, a \sin \theta_i)$ for $i = 1, 2, 3$ represent the vertices of an equilateral triangle inscribed in a circle $x^2 + y^2 = a^2$,then:

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

If the lines $3x - 4y + 4 = 0$ and $6x - 8y - 7 = 0$ are tangent to a circle,then the radius of the circle is

The position of the point $(0.1, 3.1)$ with respect to the circle $x^2 + y^2 - 2x - 4y + 3 = 0$ is:

Consider the circles $x^2 + (y - 1)^2 = 9$ and $(x - 1)^2 + y^2 = 25$. Which of the following is true?

In the figure shown,the radius of circle $C_1$ is $r$ and that of $C_2$ is $\frac{r}{2}$,where $r = \frac{1}{3} PQ$. Then the length of $AB$ is (where $P$ and $Q$ are the centers of $C_1$ and $C_2$ respectively).