Question

The point on the \( y \)-axis which is equidistant from the points \( (5, -2) \) and \( (-3, 2) \) is:

• A. \( (0, 3) \)
• B. \( (-2, 0) \)
• C. \( (0, -2) \)
• D. \( (2, 2) \)

Hint:

For an equidistant point on the \( y \)-axis, use the distance formula and set distances equal to solve for \( y \).

Answer 1

The Correct Option is C

Step 1: Let the required point be \( (0, y) \) Since the point lies on the \( y \)-axis, its coordinates are \( (0, y) \). Step 2: Use the distance formula The distance of \( (0, y) \) from \( (5, -2) \): \[ d_1 = \sqrt{(0 - 5)^2 + (y + 2)^2} = \sqrt{25 + (y + 2)^2} \] The distance of \( (0, y) \) from \( (-3, 2) \): \[ d_2 = \sqrt{(0 + 3)^2 + (y - 2)^2} = \sqrt{9 + (y - 2)^2} \] Step 3: Equate distances and solve for \( y \) \[ \sqrt{25 + (y+2)^2} = \sqrt{9 + (y-2)^2} \] Squaring both sides: \[ 25 + (y+2)^2 = 9 + (y-2)^2 \] Expanding: \[ 25 + y^2 + 4y + 4 = 9 + y^2 - 4y + 4 \] Cancel \( y^2 \) on both sides: \[ 25 + 4y + 4 = 9 - 4y + 4 \] \[ 29 + 4y = 13 - 4y \] \[ 8y = -16 \] \[ y = -2 \] Thus, the correct answer is \( (0, -2) \).