1. **Problem Statement:** We have a triangular plot with teams Alpha, Beta, and Charlie at its corners. The area of the triangle is 37 sq units. Charlie lies on the positive X-axis. The secret room coordinates are given by the adjoint matrix elements $a_{12}$ and $a_{21}$ of the matrix $\begin{bmatrix}3 & 8 \\ 11 & 10\end{bmatrix}$. (i) Find the x-coordinate of Charlie using the determinant method. (ii) Find the y-coordinate of the medical centre on the line joining Alpha and Beta where the x-coordinate is 5, using the determinant method. --- 2. **Recall the adjoint matrix:** For a 2x2 matrix $A = \begin{bmatrix}a & b \\ c & d\end{bmatrix}$, the adjoint is: $$\text{adj}(A) = \begin{bmatrix}d & -b \\ -c & a\end{bmatrix}$$ 3. **Calculate the adjoint of the given matrix:** Given $A = \begin{bmatrix}3 & 8 \\ 11 & 10\end{bmatrix}$, $$\text{adj}(A) = \begin{bmatrix}10 & -8 \\ -11 & 3\end{bmatrix}$$ So, $a_{12} = -8$ and $a_{21} = -11$. 4. **Coordinates of the secret room:** $$ (x, y) = (a_{12}, a_{21}) = (-8, -11) $$ 5. **Find the x-coordinate of Charlie:** Since Charlie lies on the positive X-axis, its y-coordinate is 0. The area of the triangle with vertices Alpha $(x_1, y_1)$, Beta $(x_2, y_2)$, and Charlie $(x_3, 0)$ is given by: $$\text{Area} = \frac{1}{2} \left| x_1(y_2 - 0) + x_2(0 - y_1) + x_3(y_1 - y_2) \right|$$ Given area = 37, so: $$37 = \frac{1}{2} |x_1 y_2 - x_2 y_1 + x_3(y_1 - y_2)|$$ Rearranged: $$74 = |x_1 y_2 - x_2 y_1 + x_3(y_1 - y_2)|$$ 6. **Using the secret room coordinates as Alpha and Beta:** Assuming Alpha = $(-8, -11)$ and Beta = $(3, 8)$ (from the original matrix first row and second column), and Charlie = $(x_3, 0)$. Calculate: $$x_1 y_2 - x_2 y_1 = (-8)(8) - (3)(-11) = -64 + 33 = -31$$ So: $$74 = |-31 + x_3(-11 - 8)| = |-31 + x_3(-19)| = |-31 - 19 x_3|$$ 7. **Solve for $x_3$:** Two cases: - $-31 - 19 x_3 = 74$ gives: $$-19 x_3 = 105 \Rightarrow x_3 = -\frac{105}{19} \approx -5.53$$ - $-31 - 19 x_3 = -74$ gives: $$-19 x_3 = -43 \Rightarrow x_3 = \frac{43}{19} \approx 2.26$$ Since Charlie is on the positive X-axis, $x_3 = 2.26$. --- 8. **Find the y-coordinate of the medical centre on line Alpha-Beta with x=5:** Let Alpha = $(x_1, y_1) = (-8, -11)$, Beta = $(x_2, y_2) = (3, 8)$, and Medical centre = $(5, y)$. The point lies on the line joining Alpha and Beta, so the determinant of the matrix formed by these points is zero: $$\begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ 5 & y & 1 \end{vmatrix} = 0$$ Expanding the determinant: $$x_1(y_2 - y) - y_1(x_2 - 5) + 1(x_2 y - 5 y_2) = 0$$ Substitute values: $$-8(8 - y) - (-11)(3 - 5) + (3 y - 5 \times 8) = 0$$ Simplify: $$-8(8 - y) + 11( -2) + 3 y - 40 = 0$$ $$-64 + 8 y - 22 + 3 y - 40 = 0$$ $$8 y + 3 y - 64 - 22 - 40 = 0$$ $$11 y - 126 = 0$$ $$11 y = 126 \Rightarrow y = \frac{126}{11} \approx 11.45$$ 9. **Reason:** The determinant method ensures the three points are collinear (area zero), confirming the medical centre lies exactly on the line joining Alpha and Beta. --- **Final answers:** - (i) The x-coordinate of Charlie is approximately $2.26$. - (ii) The y-coordinate of the medical centre is approximately $11.45$.