Question

Find the coordinates of a point on the line x + y = 5 which is equidistant from (6, 4) and (5, 2).

Answer 1

A point lies on the line x + y = 5. We can express the y-coordinate in terms of  x:

x + y = 5

y = 5 − x

Thus, any point on this line has the form P(x, 5 − x)

The distance between two points (x₁, y₁) and (x₂, y₂) is given by distance formula

= `sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2)`

Since the point is equidistant from A(6, 4) and B(5, 2), the distances PA and PB are equal (PA = PB).

Use the equidistant condition:

`sqrt((x - 6)^2 + (y - 4)^2) = sqrt((x - 5)^2 + (y - 2)^2)`

(x − 6)² + (y − 4)² = (x − 5)² + (y − 2)²

(x − 6)² + ((5 − x) − 4)² = (x − 5)² + ((5 − x) − 2)²

(x − 6)² + (1 − x) )² = (x − 5)² + (3 − x)²

(x² − 12x + 36) + (1 − 2x + x²) = (x² − 10x + 25) + (9 − 6x + x²)

2x² − 14x + 37 = 2x² − 16x + 34

−14x + 16x = 34 − 37

2x = −3

x = −`3/2`

y = 5 − x

= `5 - (-3/2)`

= `(10 + 3)/2`

= `13/2`

The coordinates of the point are (x, y) is `(-3/2, 13/2)`.