To find the coordinates of point M that divides the line segment AB in the ratio of 1:5, we can use the section formula.
Given:
- A(1, 1)
- B(7, -2)
- The ratio \( k_1:k_2 = 1:5 \)
The section formula for dividing a line segment in the ratio \( m:n \) is given by:
\[ M\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right) \]
In our case, \( m = 1 \) and \( n = 5 \):
\[ x_1 = 1, \quad y_1 = 1 \quad (coordinates \ of \ A) \] \[ x_2 = 7, \quad y_2 = -2 \quad (coordinates \ of \ B) \]
Now substituting these values into the section formula:
\[ M\left(\frac{1 \cdot 7 + 5 \cdot 1}{1+5}, \frac{1 \cdot (-2) + 5 \cdot 1}{1+5}\right) \]
Calculating the x-coordinate of M:
\[ M_x = \frac{7 + 5}{6} = \frac{12}{6} = 2 \]
Calculating the y-coordinate of M:
\[ M_y = \frac{-2 + 5}{6} = \frac{3}{6} = 0.5 \]
Therefore, the coordinates of point M are:
\[ M(2, 0.5) \]
So, the final answer is:
\[ M(2.0, 0.5) \]