The co-ordinates of the foot of perpendicular drawn from origin to a line are `(2,3)`. Find the equation of the line.

Answer

To find the equation of the line given the coordinates of the foot of the perpendicular from the origin to the line, we can follow these steps:

### Step 1: Identify the coordinates of the foot of the perpendicular
The coordinates of the foot of the perpendicular from the origin (0, 0) to the line are given as (2, 3).

### Step 2: Determine the slope of the line from the origin to the foot of the perpendicular
The slope (m) of the line segment from the origin (0, 0) to the point (2, 3) can be calculated using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{3 - 0}{2 - 0} = \frac{3}{2} \]

### Step 3: Write the equation of the line from the origin to the foot of the perpendicular
Using the point-slope form of the equation of a line, which is given by:
\[ y - y_1 = m(x - x_1) \]
we can substitute \(m = \frac{3}{2}\), \(x_1 = 0\), and \(y_1 = 0\):
\[ y - 0 = \frac{3}{2}(x - 0) \implies y = \frac{3}{2}x \]

### Step 4: Convert the equation to standard form
To convert \(y = \frac{3}{2}x\) to standard form \(Ax + By + C = 0\), we can rearrange it:
\[ -\frac{3}{2}x + y = 0 \implies 3x - 2y = 0 \]

### Step 5: Find the equation of the line perpendicular to this line
For a line perpendicular to another line, the slopes are negative reciprocals. The slope of the line we found is \(\frac{3}{2}\), so the slope of the perpendicular line will be:
\[ -\frac{1}{\frac{3}{2}} = -\frac{2}{3} \]

### Step 6: Use the foot of the perpendicular to write the equation of the perpendicular line
Using the point-slope form again for the line with slope \(-\frac{2}{3}\) passing through the point (2, 3):
\[ y - 3 = -\frac{2}{3}(x - 2) \]

### Step 7: Simplify the equation
Distributing the slope:
\[ y - 3 = -\frac{2}{3}x + \frac{4}{3} \]
Adding 3 to both sides:
\[ y = -\frac{2}{3}x + \frac{4}{3} + 3 \]
Converting 3 to a fraction:
\[ y = -\frac{2}{3}x + \frac{4}{3} + \frac{9}{3} = -\frac{2}{3}x + \frac{13}{3} \]

### Step 8: Convert to standard form
Multiplying through by 3 to eliminate the fraction:
\[ 3y = -2x + 13 \implies 2x + 3y - 13 = 0 \]

### Final Answer
The equation of the line is:
\[ 2x + 3y - 13 = 0 \]

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