Lines 2x - by + 5 = 0 and ax + 3y = 2 are parallel. Find the relation connecting a and b.

Converting 2x - by + 5 = 0 in the form y = mx + c we get,

⇒ 2x - by + 5 = 0

⇒ by = 2x + 5

⇒ y = $\dfrac{2}{b}x + \dfrac{5}{b}$

Comparing, we get slope of this line = m₁ = $\dfrac{2}{b}$.

Converting ax + 3y = 2 in the form y = mx + c we get,

⇒ ax + 3y = 2

⇒ 3y = -ax + 2

⇒ y = $-\dfrac{a}{3}x + \dfrac{2}{3}$

Comparing, we get slope of this line = m₂ = -$\dfrac{a}{3}$

Given, two lines are parallel so their slopes will be equal,

m₁ = m₂

$\Rightarrow \dfrac{2}{b} = -\dfrac{a}{3} \\[1em] \Rightarrow 3 \times 2 = -a \times b \\[1em] \Rightarrow 6 = -ab \\[1em] \Rightarrow ab + 6 = 0 \\[1em] \Rightarrow ab = -6.$

Hence, the relation connecting a and b is ab = -6.