Find the length of perpendicular drawn from origin to the line `12x-5y=26`.

Answer

To find the length of the perpendicular drawn from the origin to the line given by the equation \(12x - 5y = 26\), we can use the distance formula from a point to a line. The formula is:

\[ d = \frac{|ax_1 + by_1 + c|}{\sqrt{a^2 + b^2}} \]

where \(ax + by + c = 0\) is the line equation, and \((x_1, y_1)\) is the point from which we want to find the distance (in this case, the origin \((0, 0)\)).

### Step 1: Rewrite the line equation in the standard form
The given line equation is \(12x - 5y = 26\). We can rewrite it in the standard form \(ax + by + c = 0\):

\[ 12x - 5y - 26 = 0 \]

Here, we identify:
- \(a = 12\)
- \(b = -5\)
- \(c = -26\)

### Step 2: Substitute the values into the distance formula
Now we will substitute \(x_1 = 0\) and \(y_1 = 0\) (the coordinates of the origin) into the distance formula:

\[ d = \frac{|12(0) + (-5)(0) - 26|}{\sqrt{12^2 + (-5)^2}} \]

### Step 3: Simplify the numerator
Calculating the numerator:

\[ d = \frac{|0 + 0 - 26|}{\sqrt{12^2 + (-5)^2}} = \frac{|-26|}{\sqrt{12^2 + 5^2}} = \frac{26}{\sqrt{144 + 25}} = \frac{26}{\sqrt{169}} \]

### Step 4: Calculate the denominator
Now we calculate the denominator:

\[ \sqrt{169} = 13 \]

### Step 5: Final calculation of distance
Now substituting back into the formula gives:

\[ d = \frac{26}{13} = 2 \]

### Conclusion
Thus, the length of the perpendicular drawn from the origin to the line \(12x - 5y = 26\) is \(2\) units.

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