Lines 2x - by + 7 = 0 and ax - 2y - 7 = 0 are perpendicular to each other.

Statement 1: $\dfrac{2}{a} = \dfrac{b}{2}$.

Statement 2: Slope of line 2x - by + 7 = 0 is $\dfrac{2}{b}$ and slope of line ax - 2y - 7 = 0 is $\dfrac{a}{2}$.

1. Both the statement are true.

2. Both the statement are false.

3. Statement 1 is true, and statement 2 is false.

4. Statement 1 is false, and statement 2 is true.

Given equation of line 2x - by + 7 = 0 and ax - 2y - 7 = 0

Converting the equation of first line in slope-intercept form (y = mx + c), we get :

⇒ 2x - by + 7 = 0

⇒ by = 2x + 7

⇒ y = $\dfrac{2}{b}x + \dfrac{7}{b}$

∴ Slope of the line (m₁) = $\dfrac{2}{b}$

Converting the equation of second line in slope-intercept form y = mx + c, we get :

⇒ ax - 2y - 7 = 0

⇒ 2y = ax - 7

⇒ y = $\dfrac{a}{2}x - \dfrac{7}{2}$

∴ Slope of the line (m₂) = $\dfrac{a}{2}$

So, statement 2 is true.

We know that,

The two lines are perpendicular if product of their slopes is -1.

$\Rightarrow m$1 \times m2 = -1\\[1em] \Rightarrow \dfrac{2}{b} \times \dfrac{a}{2} = -1\\[1em] \Rightarrow \dfrac{2}{b} = -1 \times \dfrac{2}{a}\\[1em] \Rightarrow \dfrac{2}{b} = -\dfrac{2}{a}\\[1em]

So, statement 1 is false.

Hence, option 4 is the correct option.