Find the equation to the straight line drawn at right angles to the straight line `(x)/(a)- (y)/(b)=1` through the point where it meets the axis of x.

Answer

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

### Step 1: Find the point where the given line meets the x-axis.
To find the x-intercept, we set \(y = 0\) in the equation of the line:
\[ \frac{x}{a} - \frac{0}{b} = 1 \implies \frac{x}{a} = 1 \implies x = a. \]
Thus, the point where the line meets the x-axis is \((a, 0)\).

**Hint:** To find the x-intercept of a line, set \(y = 0\) in the line's equation.

### Step 2: Determine the slope of the given line.
Rearranging the equation \(\frac{x}{a} - \frac{y}{b} = 1\) into slope-intercept form \(y = mx + c\):
\[ \frac{y}{b} = \frac{x}{a} - 1 \implies y = \frac{b}{a}x - b. \]
From this, we can see that the slope \(m_1\) of the given line is \(\frac{b}{a}\).

**Hint:** To find the slope from the standard form of a line, rearrange it into \(y = mx + c\).

### Step 3: Find the slope of the line perpendicular to the given line.
The slope \(m_2\) of a line perpendicular to another line is given by the negative reciprocal of the original slope:
\[ m_2 = -\frac{1}{m_1} = -\frac{1}{\frac{b}{a}} = -\frac{a}{b}. \]

**Hint:** The slopes of two perpendicular lines are negative reciprocals of each other.

### Step 4: Use the point-slope form to write the equation of the perpendicular line.
Using the point \((a, 0)\) and the slope \(m_2 = -\frac{a}{b}\), we can use the point-slope form of the line:
\[ y - y_1 = m(x - x_1) \implies y - 0 = -\frac{a}{b}(x - a). \]
This simplifies to:
\[ y = -\frac{a}{b}(x - a) \implies y = -\frac{a}{b}x + \frac{a^2}{b}. \]

**Hint:** The point-slope form of a line is useful when you know a point on the line and its slope.

### Step 5: Rearrange the equation into standard form.
To express the equation in standard form, we can rearrange it:
\[ \frac{a}{b}x + y - \frac{a^2}{b} = 0 \implies ay + bx - a^2 = 0. \]

**Hint:** To convert to standard form, move all terms to one side of the equation.

### Final Answer:
The equation of the straight line that is perpendicular to the given line and passes through the x-axis is:
\[ ay + bx - a^2 = 0. \]