1. **Problem Statement:** Find the ratio in which the point $(a,3)$ divides the line segment joining the points $(11,-2)$ and $(3,6)$, and calculate the value of $a$. 2. **Step 1: Use the section formula.** If a point divides the line segment joining $A(x_1,y_1)$ and $B(x_2,y_2)$ in the ratio $m:n$, then the coordinates of the point are given by: $$\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right)$$ 3. **Step 2: Given points** $A = (11,-2)$, $B = (3,6)$ and point dividing them is $(a,3)$. 4. **Step 3: Set up equations using $y$-coordinate.** $$3 = \frac{m \times 6 + n \times (-2)}{m+n} = \frac{6m - 2n}{m+n}$$ Multiply both sides by $m+n$: $$3(m+n) = 6m - 2n$$ $$3m + 3n = 6m - 2n$$ Bring terms to one side: $$3m + 3n - 6m + 2n = 0 \Rightarrow -3m + 5n = 0$$ So: $$3m = 5n \Rightarrow \frac{m}{n} = \frac{5}{3}$$ 5. **Step 4: Use ratio $m:n = 5:3$ in $x$-coordinate equation.** $$a = \frac{m \times 3 + n \times 11}{m + n} = \frac{5 \times 3 + 3 \times 11}{5 + 3} = \frac{15 + 33}{8} = \frac{48}{8} = 6$$ 6. **Answer:** The point $(6,3)$ divides the line segment joining $(11,-2)$ and $(3,6)$ in the ratio $5:3$.