Question

Line AB is perpendicular to CD. Co-ordinates of B, C and D are respectively (4, 0), (0, −1) and (4, 3). Find:

1. slope of CD
2. equation of AB

Answer 1

Given:

• B = (4, 0)
• C = (0, −1)
• D = (4, 3)

(i) The slope (m) of a line passing through (x₁, y₁)and (x₂, y₂) is:

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

For line CD, the co-ordinates are: C = (0, −1) and D = (4, 3),

⇒ Let (x₁, y₁) = (0, −1) and (x₂, y₂) = (4, 3):

`"Slope of CD"(m_(CD)) = (3 - (-1))/(4 - 0)`

`m_(CD) = (3 + 1)/4`

`m_(CD) = 4/4`

∴ m_CD = 1

(ii) Since line AB ⊥ CD, the product of their slopes is −1:

m_AB × m_CD = −1

m_AB × 1 = −1

∴ m_AB = −1

Line AB passes through point B = (4, 0) with slope m = −1.

⇒ Using the point-slope formula:

y − y₁ = m(x − x₁)

y − 0 = −1(x − 4)

y = −x + 4

⇒ Rearrange in standard form(Ax + By + C = 0):

x + y − 4 = 0

Hence, the slope of CD = 1 and the equation of line AB is x + y − 4 = 0.