If the line $y = mx + 1$ meets the circle $x^2 + y^2 + 3x = 0$ at two points equidistant from the $y$-axis and on opposite sides,then:

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:

Chords $AB$ and $CD$ of a circle intersect at a right angle at point $P$. If the lengths of $AP$, $PB$, $CP$, and $PD$ are $2$, $6$, $3$, and $4$ units respectively, then the radius of the circle is:

The line $4x - 3y + 15 = 0$ intersects the circle $x^2 + y^2 - 6x - 8y = 0$ at two points $A$ and $B$. The maximum area of $\Delta ABC$,where $C$ is a point on the circumference of the circle,will be - .............. $sq. \ units$.

Let the line $x - y = 4$ intersect the circle $C : (x - 4)^2 + (y + 3)^2 = 9$ at the points $Q$ and $R$. If $P(\alpha, \beta)$ is a point on $C$ such that $PQ = PR$, then $(6\alpha + 8\beta)^2$ is equal to . . . . . .

The number of common tangents of the circles given by $x^2 + y^2 - 8x - 2y + 1 = 0$ and $x^2 + y^2 + 6x + 8y = 0$ is