Question

Find the coordinates of a point P on y-axis so that PA ≡ PB where A ≡ (−2, 4) and B ≡ (−5, −3).

Answer 1

Step 1: Let point P be (0, y)

Since the point lies on the y-axis, its x-coordinate is 0.

Step 2: Use the distance formula for PA and PB

PA = `sqrt((0 + 2)^2 + (y − 4))`

PA = `sqrt(4 + (y − 4)^2)`

PB = `sqrt((0 + 5)^2 + (y + 3)^2)`

PB = `sqrt((0 + 5)^2 + (y + 3)^2)`

PB = `sqrt(25 + (y + 3)^2)`

Step 3: Equating the distances

= `sqrt4 + (y − 4)`

= `sqrt(25 + (y + 3)^2)`

Step 4: Square both sides to eliminate square roots

4 + (y − 4)² = 25 + (y + 3)²

4 + (y² − 8y + 16) = 25 + (y² + 6y + 9)

y² − 8y + 20 = y² + 6y + 34

Step 5: Cancel y² from both sides

−8y + 20 = 6y + 34

−8y − 6y = 34 − 20

−14y = 14

y = −1

y = (0, −1)