The angle between the pair of tangents from the point $(1, 1/2)$ to the circle $x^2 + y^2 + 4x + 2y - 4 = 0$ is:

If $P$ is a point on the circle $x^{2}+y^{2}=4$,$Q$ is a point on the straight line $5x+y+2=0$ and $x-y+1=0$ is the perpendicular bisector of $PQ$,then $13$ times the sum of the abscissae of all such points $P$ is ........... .

If the length of the chord $2x+3y+k=0$ of the circle $x^2+y^2-6x-8y+9=0$ is $2\sqrt{3}$,then one of the values of $k$ is

Suppose that two chords,drawn from the point $(1, 2)$ on the circle $x^2 + y^2 + x - 3y = 0$,are bisected by the $y$-axis. If the other ends of these chords are $R$ and $S$,and the midpoint of the line segment $RS$ is $(\alpha, \beta)$,then $6(\alpha + \beta)$ is equal to:

If $x^2+y^2=25$,then $\log _5[\max (3 x+4 y)]$ is

$A$ circle touching the $x-$ axis at $(3, 0)$ and making an intercept of length $8$ on the $y-$ axis passes through the point: