Let a chord $AB$ subtend an angle of $60^{\circ}$ at the centre $C(2,3)$ of a circle $S$. If the equation of $AB$ is $x+y+1=0$,then the equation of the circle $S$ is

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 $\alpha + \beta + r =$

Consider a triangle $\Delta$ whose two sides lie on the $x$-axis and the line $x+y+1=0$. If the orthocenter of $\Delta$ is $(1,1)$,then the equation of the circle passing through the vertices of the triangle $\Delta$ is

Let a circle $C$ pass through the points $(4, 2)$ and $(0, 2)$,and its centre lie on the line $3x + 2y + 2 = 0$. Then the length of the chord of the circle $C$ whose midpoint is $(1, 2)$ is:

The area (in square units) of the circle which touches the lines $4x + 3y = 15$ and $4x + 3y = 5$ is

The set of values of $k$ for which the circle $C : 4x^{2} + 4y^{2} - 12x + 8y + k = 0$ lies inside the fourth quadrant and the point $(1, -1/3)$ lies on or inside the circle $C$ is