Find the equation of the line perpendicular to the line `x/a-y/b=1` and passing through a point at which it cuts the x-axis.

Explanation

To find the equation of the line that is perpendicular to the line given by the equation \( \frac{x}{a} - \frac{y}{b} = 1 \) and passes through the point where it intersects the x-axis, we can follow these steps:

### Step 1: Identify the given line's equation
The given line is represented as:
\[ \frac{x}{a} - \frac{y}{b} = 1 \]

### Step 2: Find the point of intersection with the x-axis
At the x-axis, the value of \( y \) is 0. Substituting \( y = 0 \) into the equation:
\[ \frac{x}{a} - \frac{0}{b} = 1 \implies \frac{x}{a} = 1 \implies x = a \]
Thus, the point of intersection with the x-axis is \( (a, 0) \).

### Step 3: Find the slope of the given line
To find the slope of the given line, we can rewrite the equation in slope-intercept form \( y = mx + c \):
\[ \frac{x}{a} - \frac{y}{b} = 1 \implies \frac{y}{b} = \frac{x}{a} - 1 \implies y = \frac{b}{a}x - b \]
From this, we see that the slope \( m \) of the given line is \( \frac{b}{a} \).

### Step 4: Find the slope of the perpendicular line
The slope of a line that is perpendicular to another line is the negative reciprocal of the original line's slope. Therefore, the slope \( m_{\perp} \) of the perpendicular line is:
\[ m_{\perp} = -\frac{1}{\frac{b}{a}} = -\frac{a}{b} \]

### Step 5: Use point-slope form to find the equation of the perpendicular line
We have the slope of the perpendicular line and a point it passes through, \( (a, 0) \). Using the point-slope form of the equation of a line:
\[ y - y_1 = m(x - x_1) \]
Substituting \( (x_1, y_1) = (a, 0) \) and \( m = -\frac{a}{b} \):
\[ y - 0 = -\frac{a}{b}(x - a) \]
This simplifies to:
\[ y = -\frac{a}{b}x + \frac{a^2}{b} \]

### Step 6: Rearranging to standard form
To express this in the standard form, we can rearrange it:
\[ \frac{a}{b}x + y = \frac{a^2}{b} \]
Multiplying through by \( b \) to eliminate the fraction:
\[ ax + by = a^2 \]

### Final Equation
Thus, the equation of the line perpendicular to the given line and passing through the point where it intersects the x-axis is:
\[ ax + by = a^2 \]