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 follow these steps:

### Step 1: Rewrite the line equation in the standard form
We start by rewriting the line equation in the standard form \(Ax + By + C = 0\).

Given the equation:
\[ 12x - 5y = 26 \]
We can rearrange it to:
\[ 12x - 5y - 26 = 0 \]
Here, \(A = 12\), \(B = -5\), and \(C = -26\).

### Step 2: Identify the point
The point from which we need to find the perpendicular distance is the origin, which has coordinates \((x_1, y_1) = (0, 0)\).

### Step 3: Use the formula for the distance from a point to a line
The formula for the distance \(d\) from a point \((x_1, y_1)\) to a line \(Ax + By + C = 0\) is given by:
\[ d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \]

### Step 4: Substitute the values into the formula
Substituting \(A = 12\), \(B = -5\), \(C = -26\), and \((x_1, y_1) = (0, 0)\) into the formula:
\[ d = \frac{|12(0) + (-5)(0) - 26|}{\sqrt{12^2 + (-5)^2}} \]
This simplifies to:
\[ d = \frac{|-26|}{\sqrt{144 + 25}} \]
\[ d = \frac{26}{\sqrt{169}} \]

### Step 5: Simplify the expression
Since \(\sqrt{169} = 13\), we have:
\[ d = \frac{26}{13} = 2 \]

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

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