A line perpendicular to the line segment joining the points A (1,0) and B (2,3) divides it at C in the ratio of 1:3. Then the equation of the line is 

To find the equation of the line perpendicular to the line segment joining points \( A(1,0) \) and \( B(2,3) \) and dividing it at \( C \) in the ratio \( 1:3 \), follow these steps:

### 1. Find the Coordinates of \( C \):

The coordinates of point \( C \), which divides the line segment \( AB \) in the ratio \( 1:3 \), can be determined using the section formula:

\[
C = \left(\frac{m_1x_2 + m_2x_1}{m_1 + m_2}, \frac{m_1y_2 + m_2y_1}{m_1 + m_2}\right)
\]

Here, \( m_1 = 1 \), \( m_2 = 3 \), \( (x_1, y_1) = (1, 0) \), and \( (x_2, y_2) = (2, 3) \).

\[
C = \left(\frac{1 \cdot 2 + 3 \cdot 1}{1 + 3}, \frac{1 \cdot 3 + 3 \cdot 0}{1 + 3}\right) = \left(\frac{2 + 3}{4}, \frac{3 + 0}{4}\right) = \left(\frac{5}{4}, \frac{3}{4}\right)
\]

So, the coordinates of \( C \) are \( \left(\frac{5}{4}, \frac{3}{4}\right) \).

### 2. Determine the Slope of \( AB \):

The slope of the line segment \( AB \) is given by:

\[
\text{slope

Learn how to find the equation of a line perpendicular to AB(1,0) and AB(2,3) that divides the segment at C in the ratio 1:3. Detailed solution includes the use of coordinate geometry and the section formula. Suitable for Grades 10-12.

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