1. **State the problem:** Given a parallelogram OABC with \(\overrightarrow{OA} = 3a\) and \(\overrightarrow{OB} = 4b\), points C, B, and X are collinear with \(CB : BX = 6 : 1\). The point Y satisfies \(\overrightarrow{CY} = 5 \overrightarrow{AX}\). Find \(\overrightarrow{OY}\) in terms of \(a\) and \(b\). 2. **Find vector \(\overrightarrow{OC}\):** Since OABC is a parallelogram, \(\overrightarrow{OC} = \overrightarrow{OA} + \overrightarrow{OB} = 3a + 4b\). 3. **Express vector \(\overrightarrow{CX}\):** Points C, B, X are collinear with ratio \(CB : BX = 6 : 1\). Treating B as division point between C and X: - Vector \(\overrightarrow{CB} = \overrightarrow{OB} - \overrightarrow{OC} = 4b - (3a + 4b) = -3a\). - Since \(CB : BX = 6 : 1\), \(\overrightarrow{BX} = \frac{1}{6} \overrightarrow{CB} = \frac{1}{6}(-3a) = -\frac{1}{2}a\). - Therefore, \(\overrightarrow{CX} = \overrightarrow{CB} + \overrightarrow{BX} = -3a - \frac{1}{2}a = -\frac{7}{2}a\). So, \(\overrightarrow{CX} = -\frac{7}{2}a\). 4. **Express vector \(\overrightarrow{AX}\):** \(\overrightarrow{AX} = \overrightarrow{OX} - \overrightarrow{OA}\) where \(\overrightarrow{OX} = \overrightarrow{OC} + \overrightarrow{CX} = (3a + 4b) - \frac{7}{2}a = (3 - \frac{7}{2})a + 4b = -\frac{1}{2}a + 4b\). Hence, $$\overrightarrow{AX} = \overrightarrow{OX} - \overrightarrow{OA} = \left(-\frac{1}{2}a + 4b\right) - 3a = -\frac{1}{2}a + 4b - 3a = -\frac{7}{2}a + 4b$$ 5. **Find vector \(\overrightarrow{CY}\):** Given \(\overrightarrow{CY} = 5 \overrightarrow{AX} = 5 \left(-\frac{7}{2}a + 4b\right) = -\frac{35}{2}a + 20b\). 6. **Find vector \(\overrightarrow{OY}\):** Since \(\overrightarrow{OY} = \overrightarrow{OC} + \overrightarrow{CY} = (3a + 4b) + \left(-\frac{35}{2}a + 20b\right) = \left(3 - \frac{35}{2}\right)a + (4 + 20)b = -\frac{29}{2}a + 24b\). **Final answer:** $$\boxed{\overrightarrow{OY} = -\frac{29}{2}a + 24b}$$