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

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

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

⇒ $\dfrac{b}{a} \times m_1 = -1$

⇒ m₁ = $-\dfrac{a}{b}$.

Equation of line through P 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.