A point P(x,y) moves so that the sum of its distances from point (4,2) and `(-2,2)` is 8. If the locus of P is an ellipse then its length of semi-major axis is

Options

A. 8
B. 2
C. 4
D. None of these

Answer

To solve the problem, we need to find the semi-major axis of the ellipse defined by the condition that the sum of distances from point \( P(x, y) \) to the points \( (4, 2) \) and \( (-2, 2) \) is equal to 8.

### Step-by-Step Solution:

1. **Identify the Points and the Condition**:
- We have two fixed points: \( A(4, 2) \) and \( B(-2, 2) \).
- The condition states that the sum of the distances from point \( P(x, y) \) to these two points is equal to 8:
\[ PA + PB = 8 \]

2. **Use the Distance Formula**:
- The distance from point \( P(x, y) \) to point \( A(4, 2) \) is:
\[ PA = \sqrt{(x - 4)^2 + (y - 2)^2} \]
- The distance from point \( P(x, y) \) to point \( B(-2, 2) \) is:
\[ PB = \sqrt{(x + 2)^2 + (y - 2)^2} \]

3. **Set Up the Equation**:
- According to the condition:
\[ \sqrt{(x - 4)^2 + (y - 2)^2} + \sqrt{(x + 2)^2 + (y - 2)^2} = 8 \]

4. **Square Both Sides**:
- To eliminate the square roots, we can square both sides. However, it’s easier to rearrange and isolate one of the square roots:
\[ \sqrt{(x - 4)^2 + (y - 2)^2} = 8 - \sqrt{(x + 2)^2 + (y - 2)^2} \]
- Now square both sides:
\[ (x - 4)^2 + (y - 2)^2 = (8 - \sqrt{(x + 2)^2 + (y - 2)^2})^2 \]

5. **Expand and Simplify**:
- Expanding the right-hand side:
\[ (8 - \sqrt{(x + 2)^2 + (y - 2)^2})^2 = 64 - 16\sqrt{(x + 2)^2 + (y - 2)^2} + (x + 2)^2 + (y - 2)^2 \]
- Set the two sides equal:
\[ (x - 4)^2 + (y - 2)^2 = 64 - 16\sqrt{(x + 2)^2 + (y - 2)^2} + (x + 2)^2 + (y - 2)^2 \]

6. **Rearranging Terms**:
- Move all terms involving square roots to one side and combine like terms.

7. **Further Simplification**:
- After some algebraic manipulation, we will arrive at a standard form of the ellipse equation, typically:
\[ \frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1 \]
- Here, \( a \) is the semi-major axis and \( b \) is the semi-minor axis.

8. **Identify the Length of the Semi-Major Axis**:
- From the derived equation, we can identify the value of \( a \) (the semi-major axis). In this case, we find that:
\[ a = 4 \]

### Final Answer:
The length of the semi-major axis is \( \boxed{4} \).