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 $2(x_1 + x_2 + x_3) + 3(y_1 + y_2 + y_3)$.

Tangent $L_1 \equiv 3x - 4y - 8 = 0$ and the chord $L_2 \equiv x + y - 1 = 0$ are at a distance of $2$ and $\sqrt{2}$ units respectively from the centre of a circle $S$. $(h, k)$ is the centre of $S$ such that $h^2 + k^2 = 13$. If the midpoint of the chord $L_2 = 0$ is $(\alpha, \beta)$ and the radius of the circle is $r$,then $\alpha + \beta + r =$

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

Let $PQ$ and $RS$ be the tangents at the extremities of the diameter $PR$ of a circle of radius $r$. If $PS$ and $RQ$ intersect at a point $X$ on the circumference of the circle,then $(PQ \cdot RS)$ is equal to

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

$A$ circle is drawn in a sector of a larger circle of radius $r$,as shown in the figure. The smaller circle is tangent to the two bounding radii and the arc of the sector. The radius of the small circle is