Question

Point B has coordinates ​(​1,2​). The​ x-coordinate of point A is -8. The distance between point A and point B is 15 units. What are the possible coordinates of point​ A?

Answer 1

The possible coordinates of point A are
`A_(1) (x,y) = (-8, 14)` and
`A_(2) (x,y) = (-8, -10)`, respectively.

Explanation:

From Analytical Geometry, we have the Equation of the Distance of a Line Segment between two points:

`l_(AB) = \sqrt{(x_(B)-x_(A))^(2) + (y_(B)-y_(A))^(2)}` (1)

Where:

`l_(AB)` - Length of the line segment AB.

`x_(A), x_(B)` - x-coordinates of points A and B.

`y_(A), y_(B)` - y-coordinates of points A and B.

If we know that
`l_(AB) = 15`,
`x_(A) = -8`,
`x_(B) = 1` and
`y_(B) = 2`, then the possible coordinates of point A is:

`\sqrt{(1+8)^(2)+(2-y_(A))^(2)} = 15`

`81 + (2-y_(A))^(2) = 225`

`(2-y_(A))^(2) = 144`

`2-y_(A) = \pm 12`

There are two possible solutions:

1)
`2-y_(A) = -12`

`y_(A) = 14`

2)
`2 - y_(A) = 12`

`y_(A) = -10`

The possible coordinates of point A are
`A_(1) (x,y) = (-8, 14)` and
`A_(2) (x,y) = (-8, -10)`, respectively.