The length of the perpendicular drawn from orgin upon the straight line `x/3 - y/4 =1` is

Options

A. `2(2)/(5)`
B. `3(1)/(5)`
C. `4 (2)/(5)`
D. `3 (2)/(5)`

Answer

To find the length of the perpendicular drawn from the origin to the straight line given by the equation \( \frac{x}{3} - \frac{y}{4} = 1 \), we can follow these steps:

### Step 1: Rewrite the line equation in standard form
We start with the equation:
\[ \frac{x}{3} - \frac{y}{4} = 1 \]
To eliminate the fractions, we can multiply through by the least common multiple (LCM) of the denominators, which is 12:
\[ 12 \left( \frac{x}{3} \right) - 12 \left( \frac{y}{4} \right) = 12 \cdot 1 \]
This simplifies to:
\[ 4x - 3y = 12 \]

### Step 2: Convert to the general form of the line equation
We can rearrange this equation into the general form \( Ax + By + C = 0 \):
\[ 4x - 3y - 12 = 0 \]
Here, \( A = 4 \), \( B = -3 \), and \( C = -12 \).

### Step 3: Use the formula for the distance from a point to a line
The formula for the distance \( d \) from a point \( (x_0, y_0) \) to the line \( Ax + By + C = 0 \) is given by:
\[ d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \]
In our case, the point is the origin \( (0, 0) \), so \( x_0 = 0 \) and \( y_0 = 0 \).

### Step 4: Substitute the values into the formula
Substituting \( A = 4 \), \( B = -3 \), \( C = -12 \), \( x_0 = 0 \), and \( y_0 = 0 \) into the formula gives:
\[ d = \frac{|4(0) + (-3)(0) - 12|}{\sqrt{4^2 + (-3)^2}} \]
This simplifies to:
\[ d = \frac{|-12|}{\sqrt{16 + 9}} = \frac{12}{\sqrt{25}} \]

### Step 5: Simplify the expression
Since \( \sqrt{25} = 5 \), we have:
\[ d = \frac{12}{5} \]

### Final Answer
The length of the perpendicular drawn from the origin to the line \( \frac{x}{3} - \frac{y}{4} = 1 \) is:
\[ \frac{12}{5} \text{ units} \]

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