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

Given: The straight lines 8px + (2 – 3p)y + 1 = 0 and

px + 8y – 7 = 0 are perpendicular to each other.

Since the lines are perpendicular to each other, product of their slopes is equal to –1.

Slope of the first line 8px + (2 – 3p)y + 1 = 0 is m1.

i.e.

Slope of the second line px + 8y – 7 = 0 is m2.

i.e.

As the lines are perpendicular to each other m1× m2 = –1.

⇒

⇒

⇒ p² = –1×(2–3p)

⇒ p² = 3p–2

⇒ p²–3p + 2 = 0

⇒ p.p–2p–p + 2 = 0

⇒ p(p–2)–1(p–2) = 0

⇒ (p–2)(p–1) = 0

⇒ p–2 = 0 ,p–1 = 0

⇒ p = 2,p = 1

Hence p = 1,2.