Does the point (3, 0) lie on the curve `3x^(2)+y^(2)-4x+7=0`?

Answer

To determine whether the point (3, 0) lies on the curve defined by the equation \(3x^2 + y^2 - 4x + 7 = 0\), we will substitute the coordinates of the point into the equation and check if the equation holds true.

### Step-by-step solution:

1. **Identify the point and the equation**:
- The point given is \( (3, 0) \).
- The equation of the curve is \( 3x^2 + y^2 - 4x + 7 = 0 \).

2. **Substitute the values of \(x\) and \(y\)**:
- Replace \(x\) with \(3\) and \(y\) with \(0\) in the equation:
\[ 3(3)^2 + (0)^2 - 4(3) + 7 = 0 \]

3. **Calculate \(3(3)^2\)**:
- First, calculate \(3^2\):
\[ 3^2 = 9 \]
- Now multiply by \(3\):
\[ 3 \times 9 = 27 \]

4. **Calculate \((0)^2\)**:
- Since \(0^2 = 0\), we have:
\[ (0)^2 = 0 \]

5. **Calculate \(-4(3)\)**:
- Multiply:
\[ -4 \times 3 = -12 \]

6. **Combine all the terms**:
- Now substitute back into the equation:
\[ 27 + 0 - 12 + 7 = 0 \]
- Simplifying this:
\[ 27 - 12 + 7 = 15 + 7 = 22 \]

7. **Check if the result equals zero**:
- Since \(22 \neq 0\), the equation does not hold true.

8. **Conclusion**:
- Therefore, the point \( (3, 0) \) does not lie on the curve defined by the equation \( 3x^2 + y^2 - 4x + 7 = 0 \).

### Final answer:
The point \( (3, 0) \) does not lie on the curve.

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