If the distance of the point P(x, y) from `A(a, 0)` is `a+x`, then `y^(2)=?`

Options

A. 2ax
B. 4ax
C. 6ax
D. 8ax

Answer

To solve the problem step by step, we start with the given information and apply the distance formula.

### Step-by-Step Solution:

1. **Identify the Points**:
- Let \( P(x, y) \) be the point we are considering.
- Let \( A(a, 0) \) be the fixed point.

2. **Distance Formula**:
The distance \( d \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Here, \( P \) corresponds to \( (x_1, y_1) \) and \( A \) corresponds to \( (x_2, y_2) \).

3. **Substituting the Coordinates**:
For our points:
- \( P(x, y) \) gives \( x_1 = x \) and \( y_1 = y \).
- \( A(a, 0) \) gives \( x_2 = a \) and \( y_2 = 0 \).

Therefore, the distance from \( P \) to \( A \) is:
\[ d = \sqrt{(a - x)^2 + (0 - y)^2} = \sqrt{(a - x)^2 + y^2} \]

4. **Setting Up the Equation**:
According to the problem, the distance \( d \) is given as \( a + x \). Thus, we can write:
\[ a + x = \sqrt{(a - x)^2 + y^2} \]

5. **Squaring Both Sides**:
To eliminate the square root, we square both sides:
\[ (a + x)^2 = (a - x)^2 + y^2 \]

6. **Expanding Both Sides**:
Using the identity \( (A + B)^2 = A^2 + B^2 + 2AB \) and \( (A - B)^2 = A^2 + B^2 - 2AB \):
\[ a^2 + 2ax + x^2 = a^2 - 2ax + x^2 + y^2 \]

7. **Simplifying the Equation**:
Cancel \( a^2 \) and \( x^2 \) from both sides:
\[ 2ax = -2ax + y^2 \]

8. **Rearranging the Terms**:
Bring \( -2ax \) to the left side:
\[ 2ax + 2ax = y^2 \]
This simplifies to:
\[ 4ax = y^2 \]

9. **Final Result**:
Thus, we find:
\[ y^2 = 4ax \]

### Conclusion:
The value of \( y^2 \) is \( 4ax \).