Question

Find the value of p for which the straight lines 8px + (2 - 3p)y + 1 = 0 and px + 8y - 7 = 0 are perpendicular to each other.

Answer 1

Given lines are 8px + (2 - 3p)y + 1 = 0 and px + 8y - 7 = 0

Let m₁ and m₂ be the slopes of the given lines

`m_1 = - ("Co - efficient of x")/("Co - efficient of y") = (- 8"p")/(2 - 3"p")`

`m_2 = - ("Co - efficient of x")/("Co - efficient of y") = (- "p")/(8)`

Since the given lines are perpendicular,

m₁m₂ = - 1

`((-8"p")/(2 - 3"p")) xx - "p"/8` = - 1

`=> (8"p"^2)/(2 - 3"p") = - 8`

⇒ 8p² = - 16 + 24p

⇒ 8p² - 24p + 16 = 0

⇒ p² - 3p + 2 = 0

⇒ p² - 2p - p + 2 = 0

⇒ p(p - 2) - 1(p - 2) = 0 

⇒ (p - 1)(p - 2) = 0

⇒ p = 1, 2