Question

A and B are two points on the X-axis and Y-axis, respectively. P(2, −1)

1. coordinates of A and B.
2. slope of AB.
3. equation of AB.

Answer 1

(i)

Since A lies on the X-axis, and B lies on the Y-axis, its coordinates are,

(x, 0) and (0, y)

Given that P(2, −1) is the midpoint of AB, we use the midpoint formula:

`((x_1 + x_2)/2, (y_1 + y_2)/2) = (x_p, y_p)`

⇒ For the x-coordinate,

`(x + 0)/2 = 2`

x = 2(2)

∴ x = 4

⇒ For the y-coordinate,

`(0 + y)/2 = -1`

y = −1(2)

∴ y = −2

Coordinates of A and B are (4, 0) and (0, −2).

(ii)

Using the slope formula with points A(4, 0) and B(0, −2),

`m = (y_2 - y_1)/(x_2 - x_1)`

= `(-2 - 0)/(0 - 4)`

= `(-2)/-4`

∴ m = `1/2`

Slope of AB is m = `1/2`.

(iii)

Using the intercept form of a line equation because the x-intercept (a = 4) and the y-intercept (b = −2):

`x/a + y/b = 1`

`x/4 + y/-2 = 1`

x − 2y = 4   ...[multiplied the entire equation by 4]

∴ x − 2y − 4 = 0

The equation of AB is x − 2y − 4 = 0 or x − 2y = 4.