Find the equation of the line which is perpendicular to the line $\dfrac{x}{a} - \dfrac{y}{b} = 1$ at the point where the given line meets y-axis.

Let A be the point where the line $\dfrac{x}{a} - \dfrac{y}{b} = 1$ meets y-axis.

So, x-co-ordinate of point A will be zero.

Substituting x = 0 in equation we get,

$\Rightarrow \dfrac{0}{a} - \dfrac{y}{b} = 1 \\[1em] \Rightarrow -\dfrac{y}{b} = 1 \\[1em] \Rightarrow y = -b$

A = (0, -b).

The given line equation is,

$\Rightarrow \dfrac{x}{a} - \dfrac{y}{b} = 1 \\[1em] \Rightarrow \dfrac{y}{b} = \dfrac{x}{a} - 1 \\[1em] \Rightarrow y= \dfrac{bx}{a} - b$

Comparing above equation with y = mx + c we get,

Slope (m) = $\dfrac{b}{a}$

Let slope of perpendicular line be m₁.

As product of slope of perpendicular lines is -1,

∴ m × m₁ = -1

⇒ $\dfrac{b}{a}$ × m₁ = -1

⇒ m₁ = $-\dfrac{a}{b}$

Equation of line through A (0, -b) and slope = $-\dfrac{a}{b}$ is :

⇒ y - y₁ = m(x - x₁)

⇒ y - (-b) = $-\dfrac{a}{b}$(x - 0)

⇒ b(y + b) = -ax

⇒ by + b² = -ax

⇒ ax + by + b² = 0

Hence, equation of required line is ax + by + b² = 0.