Question:

Step 1: Determine the domain constraints for $\sec^{-1}(x)$. For $\sec^{-1}(x)$ to be defined, we require: $$|x| \geq 1$$ Step 2: Determine the domain constraints for $\tan^{-1}\sqrt{9y^2-1}$. For the expression under the square root to be non-negative: $$9y^2 - 1 \geq 0$$ $$9y^2 \geq 1$$ $$|y| \geq \frac{1}{3}$$ Step 3: Determine the domain constraints for $\sin^{-1}(x^2+y^2)$. For $\sin^{-1}(x^2+y^2)$ to be defined, the argument must lie in $[-1, 1]$. Since $x^2 + y^2 \geq 0$, we need: $$0 \leq x^2 + y^2 \leq 1$$ Step 4: Find the intersection of all domain constraints. From Step 1: $|x| \geq 1$ implies $x^2 \geq 1$ Therefore: $x^2 + y^2 \geq 1 + y^2 \geq 1$ Combined with the constraint from Step 3 that $x^2 + y^2 \leq 1$, we must have: $$x^2 + y^2 = 1$$ This means $y^2 = 1 - x^2 = 0$, so $y = 0$ and $x = \pm 1$. Step 5: Check if the solution satisfies all constraints. When $x = \pm 1$ and $y = 0$: - The constraint $|x| \geq 1$ is satisfied ✓ - The constraint $|y| \geq \frac{1}{3}$ requires $|0| \geq \frac{1}{3}$, which is **NOT satisfied** ✗ Step 6: Conclude about the solution set. Since there are no values of $x$ and $y$ that simultaneously satisfy all three domain constraints, the domain of the left-hand side is empty. Therefore, the equation has no solution for **any value of $\lambda$**. The exhaustive set of values of $\lambda$ for which the equation has no solution is $\mathbb{R}$. **Final Answer: Option 1** — $\mathbb{R}$