1. **Problem Statement:** Given vectors $\mathbf{a} = \overrightarrow{OA}$ and $\mathbf{b} = \overrightarrow{OB}$, point $P$ lies on $AB$ such that $BA = 4BP$, and $Q$ is the midpoint of $OA$. Lines $OP$ and $BQ$ intersect at $X$. We need to express various vectors and find relations involving $h$ and $k$. 2. **Express vectors in terms of $\mathbf{a}$ and $\mathbf{b}$:** (i) Vector $\overrightarrow{AB}$: $$\overrightarrow{AB} = \overrightarrow{OB} - \overrightarrow{OA} = \mathbf{b} - \mathbf{a}$$ (ii) Vector $\overrightarrow{OP}$: Since $P$ lies on $AB$ with $BA = 4BP$, the ratio $BP : BA = 1 : 4$, so $P$ divides $AB$ in ratio $1:3$ starting from $B$ to $A$. Using section formula from $B$ to $A$: $$\overrightarrow{OP} = \overrightarrow{OB} + \frac{1}{4} \overrightarrow{BA} = \mathbf{b} + \frac{1}{4}(\mathbf{a} - \mathbf{b}) = \frac{1}{4}\mathbf{a} + \frac{3}{4}\mathbf{b}$$ (iii) Vector $\overrightarrow{BQ}$: $Q$ is midpoint of $OA$, so $$\overrightarrow{OQ} = \frac{1}{2} \mathbf{a}$$ Then, $$\overrightarrow{BQ} = \overrightarrow{OQ} - \overrightarrow{OB} = \frac{1}{2} \mathbf{a} - \mathbf{b}$$ 3. **Express $\overrightarrow{OX}$ in terms of $\mathbf{a}$, $\mathbf{b}$, and $h$ given $\overrightarrow{BX} = h \overrightarrow{BQ}$:** Write $\overrightarrow{OX}$ as: $$\overrightarrow{OX} = \overrightarrow{OB} + \overrightarrow{BX} = \mathbf{b} + h \left( \frac{1}{2} \mathbf{a} - \mathbf{b} \right) = h \frac{1}{2} \mathbf{a} + (1 - h) \mathbf{b}$$ 4. **Express $\overrightarrow{OX}$ in terms of $k$ and $\overrightarrow{OP}$ given $\overrightarrow{OX} = k \overrightarrow{OP}$:** Recall from step 2(ii): $$\overrightarrow{OP} = \frac{1}{4} \mathbf{a} + \frac{3}{4} \mathbf{b}$$ So, $$\overrightarrow{OX} = k \overrightarrow{OP} = k \left( \frac{1}{4} \mathbf{a} + \frac{3}{4} \mathbf{b} \right) = \frac{k}{4} \mathbf{a} + \frac{3k}{4} \mathbf{b}$$ 5. **Equate the two expressions for $\overrightarrow{OX}$ and solve for $h$ and $k$:** From step 3 and 4: $$h \frac{1}{2} \mathbf{a} + (1 - h) \mathbf{b} = \frac{k}{4} \mathbf{a} + \frac{3k}{4} \mathbf{b}$$ Equate coefficients of $\mathbf{a}$ and $\mathbf{b}$: - For $\mathbf{a}$: $$h \frac{1}{2} = \frac{k}{4} \implies 2h = k$$ - For $\mathbf{b}$: $$(1 - h) = \frac{3k}{4}$$ Substitute $k = 2h$ into the second equation: $$(1 - h) = \frac{3}{4} (2h) = \frac{3h}{2}$$ Multiply both sides by 2: $$2 - 2h = 3h \implies 2 = 5h \implies h = \frac{2}{5}$$ Then, $$k = 2h = 2 \times \frac{2}{5} = \frac{4}{5}$$ 6. **Express $\overrightarrow{OX}$ in terms of $\mathbf{a}$ and $\mathbf{b}$ only:** Using $h = \frac{2}{5}$ in step 3: $$\overrightarrow{OX} = \frac{2}{5} \times \frac{1}{2} \mathbf{a} + \left(1 - \frac{2}{5} \right) \mathbf{b} = \frac{1}{5} \mathbf{a} + \frac{3}{5} \mathbf{b}$$ **Final answers:** - (a)(i) $\overrightarrow{AB} = \mathbf{b} - \mathbf{a}$ - (a)(ii) $\overrightarrow{OP} = \frac{1}{4} \mathbf{a} + \frac{3}{4} \mathbf{b}$ - (a)(iii) $\overrightarrow{BQ} = \frac{1}{2} \mathbf{a} - \mathbf{b}$ - (b) $\overrightarrow{OX} = h \frac{1}{2} \mathbf{a} + (1 - h) \mathbf{b}$ - (c) $h = \frac{2}{5}$, $k = \frac{4}{5}$ - (d) $\overrightarrow{OX} = \frac{1}{5} \mathbf{a} + \frac{3}{5} \mathbf{b}$