Question

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

Answer 1

First find the point where the line meets the $$y$$-axis (put $$x = 0$$): 

$$\frac{0}{a} - \frac{y}{b} = 1$$

$$\implies -\frac{y}{b} = 1$$

$$\implies y = -b$$ 

So the point of intersection on the $$y$$-axis is $$(0, -b)$$.

Rewrite the equation to find its slope:

$$\frac{y}{b} = \frac{x}{a} - 1$$ 

$$y = \frac{b}{a}x - b$$ 

Slope of given line $$m_1 = \frac{b}{a}$$.

Slope of the perpendicular line:

$$m = -\frac{1}{m_1} = -\frac{a}{b}$$

Equation of the line passing through $$(0, -b)$$ with slope $$-\frac{a}{b}$$: 

$$y - (-b) = -\frac{a}{b}(x - 0)$$

$$y + b = -\frac{a}{b}x$$ 

$$b(y + b) = -ax$$ 

$$by + b^2 = -ax$$ 

$$ax + by + b^2 = 0$$