A line perpendicular to the line segment joining A (1,0)and B (2,3) divides it at C in the ratio of 1:5 internally. Then the equation of 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 \(1:5\) internally, we can follow these steps:

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

   The coordinates of \(C\) dividing \(AB\) in the ratio \(1:5\) 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 \(m_2 = 5\).

   \[
   C = \left( \frac{5 \cdot 1 + 1 \cdot 2}{1 + 5}, \frac{5 \cdot 0 + 1 \cdot 3}{1 + 5} \right)
   \]
   \[
   C = \left( \frac{5 + 2}{6}, \frac{0 + 3}{6} \right) = \left( \frac{7}{6}, \frac{1}{2} \right)
   \]

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 \(C\left(\frac{7}{6}, \frac{1}{2}\right)\) and the slope \(m_{\perp} = -\frac{1}{3}\):
   \[
   y - \frac{1}{2} = -\frac{1}{3}\left(x - \frac{7}{6}\right)
   \]

5. **Simplify the equation:**

   Multiply both sides by 6 to clear the fractions:
   \[
   6\left(y - \frac{1}{2}\right) = 6\left(-\frac{1}{3}\left(x - \frac{7}{6}\right)\right)
   \]
   \[
   6y - 3 = -2(x - \frac{7}{6})
   \]
   \[
   6y - 3 = -2x + \frac{14}{6}
   \]
   \[
   6y - 3 = -2x + \frac{7}{3}
   \]

   Multiply everything by 3 to clear the fraction:
   \[
   18y - 9 = -6x + 7
   \]
   \[
   6x + 18y = 16
   \]

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 \(1:5\), is:
\[
6x + 18y = 16
\]

Do you want details or have any questions? Here are five questions you might ask next:
1. How do you find the coordinates of a point dividing a line segment in a given ratio?
2. What is the distance formula between two points in a coordinate plane?
3. How can you determine if two lines are parallel or perpendicular?
4. What is the general form of the equation of a line?
5. Can you explain how to use the point-slope form of a line equation?

**Tip:** Double-check your calculations and ensure the steps are logically followed to avoid errors, especially when dealing with fractions and ratios.

Learn how to find the equation of a line perpendicular to the segment AB joining points A(1,0) and B(2,3), dividing it at C in a ratio of 1:5 internally. Step-by-step solution provided, covering coordinate geometry concepts and line equations. Suitable for Grades 9-10.

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