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 the points \(A(1,0)\) and \(B(2,3)\) and dividing it at \(C\) in the ratio of \(1:3\), we can follow these steps:

1. **Find the coordinates of point C:**

   The coordinates of \(C\) dividing \(AB\) in the ratio \(1:3\) are given by the section formula:
   \[
   C = \left( \frac{m_2 x_1 + m_1 x_2}{m_1 + m_2}, \frac{m_2 y_1 + m_1 y_2}{m_1 + m_2} \right)
   \]
   where \(A(x_1, y_1) = (1, 0)\) and \(B(x_2, y_2) = (2, 3)\), \(m_1 = 1\), and \(m_2 = 3\).

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

2. **Find the slope of the line segment \(AB\):**

   The slope of the line segment joining \(A\) and \(B\) is:
   \[
   m_{AB} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{3 - 0}{2 - 1} = 3
   \]

3. **Find the slope of the perpendicular line:**

   The slope of the line perpendicular to \(AB\) is the negative reciprocal of \(m_{AB}\):
   \[
   m_{\perp} = -\frac{1}{m_{AB}} = -\frac{1}{3}
   \]

4. **Find the equation of the perpendicular line passing through \(C\):**

   The equation of a line in slope-intercept form is given by:
   \[
   y - y_1 = m(x - x_1)
   \]
   Using point \(C\left(\frac{5}{4}, \frac{3}{4}\right)\) and the slope \(m_{\perp} = -\frac{1}{3}\):
   \[
   y - \frac{3}{4} = -\frac{1}{3}\left(x - \frac{5}{4}\right)
   \]

5. **Simplify the equation:**

   Multiply both sides by 12 to clear the fractions:
   \[
   12\left(y - \frac{3}{4}\right) = 12\left(-\frac{1}{3}\left(x - \frac{5}{4}\right)\right)
   \]
   \[
   12y - 9 = -4(x - \frac{5}{4})
   \]
   \[
   12y - 9 = -4x + 5
   \]
   \[
   12y + 4x = 14
   \]
   \[
   4x + 12y = 14
   \]

Thus, the equation of the line perpendicular to the line segment joining the points \(A\) and \(B\), dividing it at \(C\) in the ratio of \(1:3\), is:
\[
4x + 12y = 14
\]

Do you want details or have any questions? Here are five questions you might ask next:
1. How do you find the midpoint of a line segment?
2. Can you explain how to determine if two lines are perpendicular?
3. What is the distance formula for two points in a plane?
4. How do you convert the general form of a line equation to slope-intercept form?
5. Can you show an example of finding the intersection of two lines?

**Tip:** Always check your calculations for errors by plugging back into the original equation or conditions to ensure accuracy.

Learn how to find the equation of a perpendicular line dividing the line segment joining points A(1,0) and B(2,3) at a ratio of 1:3. Detailed step-by-step solution using the section formula and slope concepts.

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