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

The circle passes through $$(4,2)$$ and $$(0,2)$$, so the midpoint of these points lies on the perpendicular bisector.

Midpoint = $$(2, 2)$$. The line joining these points is horizontal ($$y = 2$$), so the perpendicular bisector is the vertical line $$x = 2$$.

The centre lies on both $$x = 2$$ and $$3x + 2y + 2 = 0$$:

$$3(2) + 2y + 2 = 0 \Rightarrow 2y = -8 \Rightarrow y = -4$$

Centre $$= (2, -4)$$.

Radius = distance from centre to $$(4,2)$$:

$$r = \sqrt{(4-2)^2 + (2+4)^2} = \sqrt{4+36} = \sqrt{40} = 2\sqrt{10}$$

For the chord with midpoint $$(1, 2)$$:

Distance from centre $$(2, -4)$$ to midpoint $$(1, 2)$$:

$$d = \sqrt{(2-1)^2 + (-4-2)^2} = \sqrt{1+36} = \sqrt{37}$$

Half-length of chord = $$\sqrt{r^2 - d^2} = \sqrt{40 - 37} = \sqrt{3}$$

Length of chord = $$2\sqrt{3}$$.

The correct answer is Option 3: $$2\sqrt{3}$$.