Question

If the point R (x, y) is equidistant from two points P (- 3, 4) and Q (2, – 1), prove that

y = x + 2.​

Answer 1

see explanation

Explanation:

Calculate the distance between (x, y) and the 2 given points and equate them.

Using the distance formula

d =
`\sqrt{(x_(2)-x_(1))^2+(y_(2)-y_(1))^2 }`

with (x₁, y₁ ) = (x, y) and (x₂, y₂ ) = P(- 3, 4)

d =
`√((-3-x)^2+(4-y)^2)`

Repeat with (x₁, y₁ ) = (x, y) and (x₂, y₂ ) = Q(2, - 1)

d =
`√((2-x)^2+(- 1- y)^2)`

Equating both distances

`√((2-x)^2+(-1-y)^2)` =
`√((-3-x)^2+(4-y)^2)`

Square both sides

(2 - x)² + (- 1- y)² = (- 3 - x)² + (4 - y)² ← expand and simplify both sides

4 - 4x + x² + 1 + 2y + y² = 9 + 6x + x² + 16 - 8y + y²

subtract x² + y² from both sides

5 - 4x + 2y = 25 + 6x - 8y ( add 8y to both sides )

5 - 4x + 10y = 25 + 6x ( subtract 5 from both sides )

- 4x + 10y = 20 + 6x ( add 4x to both sides )

10y = 10x + 20 ( divide all terms by 10 )

y = x + 2 ← as required