The point `(a,-a)` always lies on _______.

Options

A. `x+y=0`
B. `x-y=0`
C. `x=-a`
D. `y=a`

Answer

To determine the line on which the point \((a, -a)\) always lies, we need to analyze the coordinates of the point and see which line equation it satisfies.

### Step-by-Step Solution:

1. **Identify the Coordinates**: The point given is \((a, -a)\).
- Here, \(x = a\) and \(y = -a\).

2. **Check the First Option**: Let's check if the point satisfies the equation \(y = 0\).
- Substitute \(y = -a\) into the equation:
\[ -a = 0 \]
- This is only true when \(a = 0\). Thus, this option does not hold for all values of \(a\).

3. **Check the Second Option**: Now, check if the point satisfies the equation \(x - y = 0\).
- Substitute \(x = a\) and \(y = -a\):
\[ a - (-a) = 0 \implies a + a = 0 \implies 2a = 0 \]
- This is only true when \(a = 0\). Thus, this option does not hold for all values of \(a\).

4. **Check the Third Option**: Next, check if the point satisfies the equation \(x = -a\).
- Substitute \(x = a\):
\[ a = -a \]
- This is only true when \(a = 0\). Thus, this option does not hold for all values of \(a\).

5. **Check the Fourth Option**: Finally, check if the point satisfies the equation \(y = a\).
- Substitute \(y = -a\):
\[ -a = a \]
- This is only true when \(a = 0\). Thus, this option does not hold for all values of \(a\).

6. **Identify the Correct Equation**: The point \((a, -a)\) can be expressed in terms of a linear equation:
- Rearranging the coordinates gives us:
\[ y + x = 0 \implies y = -x \]
- This means that the point \((a, -a)\) always lies on the line represented by the equation \(y = -x\).

### Conclusion:
The point \((a, -a)\) always lies on the line \(y = -x\).