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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.
Solution
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
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