Explanation:

Let's break it down:

- Original Area: The area of a circle is $\pi r^2$, where $r$ is the radius.

- New Radius: If the radius is increased by 23%, the new radius is $r\times 1.23$.

- New Area: The new area will be $\pi (r\times 1.23{)}^2=\pi r^2\times 1.5129$.

- Increase in Area: The increase in area is given by $1.5129-1.00=0.5129$ which is 51.29%.

So, the area increases by 51.29%.

- Option 1: 41.61% - Incorrect, as the calculations showed 51.29%.

- Option 2: 51.29% - Correct, matches our calculated increase.

- Option 3: 27.89% - Incorrect, too low.

- Option 4: 32.85% - Incorrect, also too low.

: The correct answer is 51.29%.