Consider this question,

Suppose that $(X_1, Y_1),(X_2, Y_2), . . . ,(X_n, Y_n)$ are the coordinates of $n$ points chosen independently and uniformly at random within a circle with center $(0, 0)$ and unknown radius $r$. Obtain the MLE $\hat{r}_n$ of $r$.

My attempt:

I have thought about the question but I am not able to put it formally. This is my reasoning. $X_i$ and $Y_i$ are both $uniformly$ $distributed$ independent random variables on $(-r,r)$. Since $(X_i, Y_i)$ are the coordinates of the $i^\text{th}$ point, transforming these coordinates to polar coordinates we get for the $i^\text{th}$ point $(\theta_i,a_i)$ (say) where $\theta_i$ follows $Uniform(0,2\pi)$ and $a_i$ follows $Uniform(0,r)$ independently. Then $a_{(n)} = max(a_i)$ is the MLE of $r$.

Is this reasoning correct? How do I put it formally? And if not, how should this problem be solved?

Consider this question,

Suppose that $(X_1, Y_1),(X_2, Y_2), . . . ,(X_n, Y_n)$ are the coordinates of $n$ points chosen independently and uniformly at random within a circle with center $(0, 0)$ and unknown radius $r$. Obtain the MLE $\hat{r}_n$ of $r$.

My attempt:

I have thought about the question but I am not able to put it formally. This is my reasoning. $X_i$ and $Y_i$ are both $uniformly$ $distributed$ independent random variables on a circle of radius $r$ and center at $(0,0)$. Since $(X_i, Y_i)$ are the coordinates of the $i^\text{th}$ point, transforming these coordinates to polar coordinates we get for the $i^\text{th}$ point $(\theta_i,a_i)$ (say) where $\theta_i$ follows $Uniform(0,2\pi)$ and $a_i$ follows $Uniform(0,r)$ independently. Then $a_{(n)} = max(a_i)$ is the MLE of $r$.

Is this reasoning correct? How do I put it formally? And if not, how should this problem be solved?