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

### Related Formula Length of a chord with perpendicular distance d$d$ from the center of a circle of radius r$r$ is: textLength = 2sqrtr^2 - d^2$$\text{Length} = 2\sqrt{r^2 - d^2}$$ ### Core Logic Points A(4,2)$A(4,2)$ and B(0,2)$B(0,2)$ have the same y-coordinate, meaning chord AB$AB$ is horizontal. The perpendicular bisector of a horizontal chord is vertical.

Midpoint of AB$AB$ is M(2,2)$M(2,2)$. Thus, the vertical line passing through the center is x = 2$x = 2$. ### Step 1: Identify Center and Radius Since the center lies on the line 3x + 2y + 2 = 0$3x + 2y + 2 = 0$, substitute x = 2$x = 2$ to find the y-coordinate: 3(2) + 2y + 2 = 0 implies 2y = -8 implies y = -4$$3(2) + 2y + 2 = 0 \implies 2y = -8 \implies y = -4$$ So, Center O = (2, -4)$O = (2, -4)$. Calculate radius r$r$ using point B(0,2)$B(0,2)$: r = OB = sqrt(2 - 0)^2 + (-4 - 2)^2 = sqrt4 + 36 = sqrt40$$r = OB = \sqrt{(2 - 0)^2 + (-4 - 2)^2} = \sqrt{4 + 36} = \sqrt{40}$$ ### Step 2: Find Target Chord Length The targeted chord has a given midpoint N(1,2)$N(1,2)$. Distance from center O(2,-4)$O(2,-4)$ to N(1,2)$N(1,2)$: d = ON = sqrt(2 - 1)^2 + (-4 - 2)^2 = sqrt1 + 36 = sqrt37$$d = ON = \sqrt{(2 - 1)^2 + (-4 - 2)^2} = \sqrt{1 + 36} = \sqrt{37}$$ textLength of chord = 2sqrtr^2 - d^2 = 2sqrt40 - 37 = 2sqrt3$$\text{Length of chord} = 2\sqrt{r^2 - d^2} = 2\sqrt{40 - 37} = 2\sqrt{3}$$ ### Pattern Recognition Points sharing a coordinate define standard vertical or horizontal perpendicular configurations immediately. Always exploit geometrical configurations before jumping into standard circle equations. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 11 Mathematics: Circles