Which equation of the chord bisects the circle $x^2 + y^2 = 8x$ at the point $(4, 3)$?

The coordinates of the centre of the smallest circle passing through the origin and having $y=x+1$ as a diameter are

The equation of the circle which touches the coordinate axes and the line $\frac{x}{3} + \frac{y}{4} = 1$,and whose centre lies in the first quadrant,is ${x^2} + {y^2} - 2cx - 2cy + {c^2} = 0$,where $c$ is:

If $(m_i, \frac{1}{m_i}), i = 1, 2, 3, 4$ are concyclic points,then the value of $m_1 \cdot m_2 \cdot m_3 \cdot m_4$ 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:

The line $2x + y = 4$ intersects the $X$ and $Y$ axes at points $A$ and $B$ respectively. Let $C(p, q)$ be a point such that the lines $AC$ and $BC$ are perpendicular. Find the distance of point $C$ from the midpoint of segment $AB$.