Question:

Step 1: Identify the diameter of the circumcircle. The problem statement indicates that for the circumcircle of $\triangle TAB$, the segment $TC_3$ serves as its diameter. Step 2: Calculate the length of the diameter $TC_3$. The length of the diameter $TC_3$ is given by the formula $TC_3 = \frac{TA}{\cos\left(\frac{\theta_1 + \theta_2}{2}\right)}$. Substituting the given values $TA = 4$ and $\cos\left(\frac{\theta_1 + \theta_2}{2}\right) = \frac{1}{\sqrt{5}}$: $$TC_3 = \frac{4}{\frac{1}{\sqrt{5}}} = 4\sqrt{5}$$ Step 3: Determine the radius of the circumcircle. The radius $R$ of the circumcircle is half of its diameter. $$R = \frac{TC_3}{2} = \frac{4\sqrt{5}}{2} = 2\sqrt{5}$$ Step 4: Calculate the area of the circumcircle. The area of a circle is given by the formula $A = \pi R^2$. Substituting the calculated radius $R = 2\sqrt{5}$: $$A = \pi \left(2\sqrt{5}\right)^2 = \pi (4 \times 5) = 20\pi$$ Step 5: State the final answer. The area of the circle is $20\pi$. This matches Option 2.