Find the point on Y-axis which is equidistant from the points `(-5,2)and (9,-2).`

Answer

To find the point on the Y-axis that is equidistant from the points (-5, 2) and (9, -2), we can follow these steps:

### Step 1: Understand the point on the Y-axis
A point on the Y-axis has coordinates of the form (0, y), where x = 0.

### Step 2: Use the distance formula
The distance \( d \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
We need to find the point (0, y) that is equidistant from the points (-5, 2) and (9, -2).

### Step 3: Set up the distance equations
1. Distance from (0, y) to (-5, 2):
\[ d_1 = \sqrt{(0 - (-5))^2 + (y - 2)^2} = \sqrt{(5)^2 + (y - 2)^2} = \sqrt{25 + (y - 2)^2} \]

2. Distance from (0, y) to (9, -2):
\[ d_2 = \sqrt{(0 - 9)^2 + (y - (-2))^2} = \sqrt{(-9)^2 + (y + 2)^2} = \sqrt{81 + (y + 2)^2} \]

### Step 4: Set the distances equal to each other
Since the point (0, y) is equidistant from both points, we set \( d_1 = d_2 \):
\[ \sqrt{25 + (y - 2)^2} = \sqrt{81 + (y + 2)^2} \]

### Step 5: Square both sides to eliminate the square roots
\[ 25 + (y - 2)^2 = 81 + (y + 2)^2 \]

### Step 6: Expand both sides
1. Left side:
\[ 25 + (y^2 - 4y + 4) = y^2 - 4y + 29 \]

2. Right side:
\[ 81 + (y^2 + 4y + 4) = y^2 + 4y + 85 \]

### Step 7: Set the equation
Now we have:
\[ y^2 - 4y + 29 = y^2 + 4y + 85 \]

### Step 8: Simplify the equation
Subtract \( y^2 \) from both sides:
\[ -4y + 29 = 4y + 85 \]
Now, move all terms involving \( y \) to one side and constant terms to the other:
\[ -4y - 4y = 85 - 29 \]
\[ -8y = 56 \]

### Step 9: Solve for \( y \)
Divide both sides by -8:
\[ y = -7 \]

### Step 10: Write the point
The point on the Y-axis is:
\[ (0, -7) \]

### Final Answer:
The point on the Y-axis which is equidistant from the points (-5, 2) and (9, -2) is (0, -7).

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