If the lines 3x + y = 4, x - ay + 7 = 0 and bx + 2y + 5 = 0 form three consecutive sides of a rectangle, find the values of a and b.

Given lines are :

3x + y = 4 ….(i)

x- ay + 7 = 0 ….(ii)

bx + 2y + 5 = 0 ….(iii)

It's said that these lines form three consecutive sides of a rectangle.

So,

Lines (i) and (ii) must be perpendicular and also (ii) and (iii) will be perpendicular.

Slope of line (i) is

⇒ 3x + y = 4

⇒ y = -3x + 4.

Comparing with y = mx + c we get,

slope = m₁ = -3.

Slope of line (ii) is

⇒ x - ay + 7 = 0

⇒ ay = x + 7

⇒ y = $\dfrac{1}{a}x + \dfrac{7}{a}$

Comparing with y = mx + c we get,

slope = m₂ = $\dfrac{1}{a}$.

Slope of line (iii) is

⇒ bx + 2y + 5 = 0

⇒ 2y = -bx - 5

⇒ y = $-\dfrac{b}{2}x - \dfrac{5}{2}$

Comparing with y = mx + c we get,

slope = m₃ = $-\dfrac{b}{2}$.

Since, lines (i) and (ii) are perpendicular so,

⇒ m₁ × m₂ = -1

$\Rightarrow -3 \times \dfrac{1}{a} = -1 \\[1em] \Rightarrow a = \dfrac{-3}{-1} \\[1em] \Rightarrow a = 3.$

Since, lines (ii) and (iii) are perpendicular so,

⇒ m₂ × m₃ = -1

$\Rightarrow \dfrac{1}{a} \times -\dfrac{b}{2} = -1 \\[1em] \Rightarrow -\dfrac{b}{2a} = -1 \\[1em] \Rightarrow b = 2a \\[1em] \Rightarrow b = 2(3) \\[1em] \Rightarrow b = 6.$

Thus, the value of a is 3 and the value of b is 6.