If A and B are two fixed points and P is a variable point such that ` PA + PB = 4`, the locus of P is

Options

A. A parabola
B. An ellipse
C. hy hyperbola
D. none of these

Answer

To solve the problem, we need to determine the locus of a variable point \( P \) such that the sum of its distances from two fixed points \( A \) and \( B \) is constant. Here are the steps to arrive at the solution:

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have two fixed points \( A \) and \( B \) in a plane, and a variable point \( P \) such that the sum of the distances from \( P \) to \( A \) and from \( P \) to \( B \) is equal to a constant value, which is given as \( PA + PB = 4 \).

2. **Identifying the Shape**:
The condition \( PA + PB = k \) (where \( k \) is a constant) is a defining property of an ellipse. In this case, since \( k = 4 \), we can conclude that the locus of point \( P \) is an ellipse.

3. **Setting Up the Ellipse**:
For an ellipse, the two fixed points \( A \) and \( B \) are known as the foci. The sum of the distances from any point on the ellipse to these two foci is constant.

4. **Conclusion**:
Since we have established that the sum of distances from \( P \) to \( A \) and \( B \) is constant and equal to 4, we conclude that the locus of the point \( P \) is indeed an ellipse.

### Final Answer:
The locus of point \( P \) is an **ellipse**.