Is the circle `x ^(2) +y ^(2) = r ^(2)` symmetrical about the line `y=x` ?

Answer

To determine if the circle given by the equation \( x^2 + y^2 = r^2 \) is symmetric about the line \( y = x \), we can follow these steps:

### Step 1: Understand the symmetry condition
A curve is symmetric about the line \( y = x \) if swapping \( x \) and \( y \) in its equation results in the same equation.

### Step 2: Write down the original equation
The equation of the circle is:
\[ x^2 + y^2 = r^2 \]

### Step 3: Swap \( x \) and \( y \)
To check for symmetry about the line \( y = x \), we replace \( x \) with \( y \) and \( y \) with \( x \):
\[ y^2 + x^2 = r^2 \]

### Step 4: Simplify the swapped equation
The equation after swapping becomes:
\[ x^2 + y^2 = r^2 \]
This is the same as the original equation.

### Step 5: Conclusion
Since the equation remains unchanged after swapping \( x \) and \( y \), we conclude that the circle \( x^2 + y^2 = r^2 \) is symmetric about the line \( y = x \).

Thus, the answer is **Yes**.

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