A (- 2, 4) and B (- 5, - 3) are two points. The coordinates of a point P on Y axis such that PA = PB, are
(a) (3, 4)
(b) (0, 9)
(c)(9, 0)
(d)(0, - 1)

Options

A. (3, 4)
B. (0, 9)
C. (9, 0)
D. (0, - 1)

Answer

To find the coordinates of point P on the Y-axis such that the distances PA and PB are equal, we can follow these steps:

### Step 1: Understand the coordinates of points A and B
- Point A has coordinates (-2, 4).
- Point B has coordinates (-5, -3).
- Point P is on the Y-axis, which means its coordinates can be represented as (0, y).

### Step 2: Use the distance formula
The distance between two points (x1, y1) and (x2, y2) is given by the formula:
\[ d = \sqrt{(x2 - x1)^2 + (y2 - y1)^2} \]
We need to find distances PA and PB.

### Step 3: Calculate distance PA
Using point A (-2, 4) and point P (0, y):
\[ PA = \sqrt{(0 - (-2))^2 + (y - 4)^2} = \sqrt{(2)^2 + (y - 4)^2} = \sqrt{4 + (y - 4)^2} \]

### Step 4: Calculate distance PB
Using point B (-5, -3) and point P (0, y):
\[ PB = \sqrt{(0 - (-5))^2 + (y - (-3))^2} = \sqrt{(5)^2 + (y + 3)^2} = \sqrt{25 + (y + 3)^2} \]

### Step 5: Set the distances equal
Since PA = PB, we can set the two expressions equal to each other:
\[ \sqrt{4 + (y - 4)^2} = \sqrt{25 + (y + 3)^2} \]

### Step 6: Square both sides to eliminate the square roots
Squaring both sides gives:
\[ 4 + (y - 4)^2 = 25 + (y + 3)^2 \]

### Step 7: Expand both sides
Expanding the left side:
\[ 4 + (y^2 - 8y + 16) = y^2 - 8y + 20 \]
Expanding the right side:
\[ 25 + (y^2 + 6y + 9) = y^2 + 6y + 34 \]

### Step 8: Set the expanded equations equal
Now we have:
\[ y^2 - 8y + 20 = y^2 + 6y + 34 \]

### Step 9: Simplify the equation
Subtract \(y^2\) from both sides:
\[ -8y + 20 = 6y + 34 \]
Rearranging gives:
\[ -8y - 6y = 34 - 20 \]
\[ -14y = 14 \]

### Step 10: Solve for y
Dividing both sides by -14:
\[ y = -1 \]

### Step 11: Write the coordinates of point P
Since point P is on the Y-axis, its coordinates are:
\[ P(0, -1) \]

### Conclusion
The coordinates of point P such that PA = PB are (0, -1), which corresponds to option (d).

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