The point on the axis of y which its equidistant from (-1, 2) and (3, 4), is

Options

A. (0, 3)
B. (0, 4)
C. (0, 5)
D. (0, -6)

Answer

To find the point on the y-axis that is equidistant from the points (-1, 2) and (3, 4), we can follow these steps:

### Step 1: Define the point on the y-axis
Since the point is on the y-axis, we can denote it as \( P(0, y) \), where \( y \) is the y-coordinate we need to find.

### Step 2: Use the distance formula
We need to find the distances from point \( P(0, y) \) to the points (-1, 2) and (3, 4). The distance formula between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

### Step 3: Calculate the distance from \( P(0, y) \) to (-1, 2)
Using the distance formula:
\[ d_1 = \sqrt{(0 - (-1))^2 + (y - 2)^2} = \sqrt{(1)^2 + (y - 2)^2} = \sqrt{1 + (y - 2)^2} \]

### Step 4: Calculate the distance from \( P(0, y) \) to (3, 4)
Similarly, we find:
\[ d_2 = \sqrt{(0 - 3)^2 + (y - 4)^2} = \sqrt{(-3)^2 + (y - 4)^2} = \sqrt{9 + (y - 4)^2} \]

### Step 5: Set the distances equal to each other
Since point \( P(0, y) \) is equidistant from both points, we set the distances equal:
\[ \sqrt{1 + (y - 2)^2} = \sqrt{9 + (y - 4)^2} \]

### Step 6: Square both sides to eliminate the square roots
Squaring both sides gives:
\[ 1 + (y - 2)^2 = 9 + (y - 4)^2 \]

### Step 7: Expand both sides
Expanding the squares:
\[ 1 + (y^2 - 4y + 4) = 9 + (y^2 - 8y + 16) \]
This simplifies to:
\[ 1 + y^2 - 4y + 4 = 9 + y^2 - 8y + 16 \]
\[ y^2 - 4y + 5 = y^2 - 8y + 25 \]

### Step 8: Cancel \( y^2 \) and simplify
Subtract \( y^2 \) from both sides:
\[ -4y + 5 = -8y + 25 \]

### Step 9: Rearrange the equation
Rearranging gives:
\[ -4y + 8y = 25 - 5 \]
\[ 4y = 20 \]

### Step 10: Solve for \( y \)
Dividing both sides by 4:
\[ y = 5 \]

### Final Step: Write the point
Thus, the point on the y-axis that is equidistant from (-1, 2) and (3, 4) is:
\[ (0, 5) \]