Find the length of the perpendicular drawn from
the point (b, a) upon the straight line `(x)/(a)-(y)/(b)= 1`.

Answer

To find the length of the perpendicular drawn from the point (b, a) to the straight line given by the equation \(\frac{x}{a} - \frac{y}{b} = 1\), we will follow these steps:

### Step 1: Rewrite the equation of the line in standard form
The given line equation is:
\[ \frac{x}{a} - \frac{y}{b} = 1 \]
To convert this into the standard form \(Ax + By + C = 0\), we can multiply through by \(ab\) to eliminate the fractions:
\[ b x - a y = a b \]
Rearranging gives:
\[ b x - a y - ab = 0 \]
Thus, we have \(A = b\), \(B = -a\), and \(C = -ab\).

### Step 2: Use the formula for the length of the perpendicular
The formula for the length \(d\) of the perpendicular from a point \((x_0, y_0)\) to the line \(Ax + By + C = 0\) is given by:
\[ d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \]
Here, our point is \((b, a)\), so \(x_0 = b\) and \(y_0 = a\).

### Step 3: Substitute the values into the formula
Substituting \(A\), \(B\), \(C\), \(x_0\), and \(y_0\) into the formula:
\[ d = \frac{|b(b) + (-a)(a) - ab|}{\sqrt{b^2 + (-a)^2}} \]
This simplifies to:
\[ d = \frac{|b^2 - a^2 - ab|}{\sqrt{b^2 + a^2}} \]

### Step 4: Final expression for the length of the perpendicular
Thus, the length of the perpendicular from the point \((b, a)\) to the line \(\frac{x}{a} - \frac{y}{b} = 1\) is:
\[ d = \frac{|b^2 - a^2 - ab|}{\sqrt{b^2 + a^2}} \]