1. **Problem statement:** Given a chord AB of length 12 units in a circle, point C lies on AB dividing it in the ratio 1:3. We need to find the length of the shortest chord passing through point C. 2. **Understanding the problem:** Point C divides AB into two segments: AC and CB such that $ \frac{AC}{CB} = \frac{1}{3} $. Since AB = 12, let AC = $x$, then CB = $3x$. So, $x + 3x = 12 \Rightarrow 4x = 12 \Rightarrow x = 3$. Thus, AC = 3 and CB = 9. 3. **Key formula:** The shortest chord through a point inside a circle is the chord perpendicular to the radius through that point. The length of a chord through point C can be found using the power of a point theorem: $$\text{Power of point C} = AC \times CB = 3 \times 9 = 27$$ 4. **Using the power of a point:** For any chord through C, if the chord length is $2d$, then the distance from C to the center $O$ and the radius $R$ satisfy: $$d = \sqrt{R^2 - OC^2}$$ But since we don't have $R$ or $OC$, we use the power of point relation: $$AC \times CB = R^2 - OC^2 = 27$$ 5. **Shortest chord through C:** The shortest chord through C is the one perpendicular to AB, and its length is: $$2 \times \sqrt{27} = 2 \times 3\sqrt{3} = 6\sqrt{3}$$ 6. **Answer:** The shortest chord passing through point C is $6\sqrt{3}$. **Final answer:** Option 4) $6\sqrt{3}$