Question:

Step 1: Express the constraint equation in standard form. We are given that $2\cos\theta - \sin\theta + 2 = 0$. Rearranging this equation: $$2\cos\theta - \sin\theta = -2$$ Step 2: Solve for $\cos\theta$ in terms of $\sin\theta$. From the constraint equation $2\cos\theta - \sin\theta = -2$, we can express $\cos\theta$ as: $$2\cos\theta = \sin\theta - 2$$ $$\cos\theta = \frac{\sin\theta - 2}{2}$$ Step 3: Use the Pythagorean identity to find $\sin\theta$. Since $\sin^2\theta + \cos^2\theta = 1$, we substitute our expression for $\cos\theta$: $$\sin^2\theta + \left(\frac{\sin\theta - 2}{2}\right)^2 = 1$$ Expanding the squared term: $$\sin^2\theta + \frac{(\sin\theta - 2)^2}{4} = 1$$ $$\sin^2\theta + \frac{\sin^2\theta - 4\sin\theta + 4}{4} = 1$$ Multiplying through by 4: $$4\sin^2\theta + \sin^2\theta - 4\sin\theta + 4 = 4$$ $$5\sin^2\theta - 4\sin\theta = 0$$ $$\sin\theta(5\sin\theta - 4) = 0$$ Therefore: $\sin\theta = 0$ or $\sin\theta = \frac{4}{5}$ Step 4: Check both cases and find the corresponding $\cos\theta$ values. **Case 1:** If $\sin\theta = 0$ From $2\cos\theta - \sin\theta = -2$: $$2\cos\theta - 0 = -2$$ $$\cos\theta = -1$$ Then: $10\cos\theta + 15\sin\theta = 10(-1) + 15(0) = -10$ This value is not among the given options. **Case 2:** If $\sin\theta = \frac{4}{5}$ From $\cos\theta = \frac{\sin\theta - 2}{2}$: $$\cos\theta = \frac{\frac{4}{5} - 2}{2} = \frac{\frac{4-10}{5}}{2} = \frac{-\frac{6}{5}}{2} = -\frac{3}{5}$$ Step 5: Calculate the required expression. With $\sin\theta = \frac{4}{5}$ and $\cos\theta = -\frac{3}{5}$: $$10\cos\theta + 15\sin\theta = 10\left(-\frac{3}{5}\right) + 15\left(\frac{4}{5}\right)$$ $$= -6 + 12 = 6$$ **Final Answer:** The value of $10\cos\theta + 15\sin\theta$ is $\boxed{6}$, which corresponds to **Option 2**.