To find the coordinates of point \( O \) that divides the line segment \( MN \) in the ratio \( 1:6 \), we can use the section formula.
Given the coordinates of points \( M \) and \( N \):
- \( M(-11, 8) \)
- \( N(73, -6) \)
Let \( O \) divide \( MN \) in the ratio \( 1:6 \). Thus, \( MO:ON = 1:6 \).
Using the section formula:
\[ O = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) \]
Where:
- \( m = 1 \) (the part of the ratio we are dividing),
- \( n = 6 \),
- \( (x_1, y_1) = (-11, 8) \) (coordinates of \( M \)),
- \( (x_2, y_2) = (73, -6) \) (coordinates of \( N \)).
Now substituting the values into the formula:
For the x-coordinate:
\[ x_O = \frac{1 \cdot 73 + 6 \cdot (-11)}{1 + 6} = \frac{73 - 66}{7} = \frac{7}{7} = 1 \]
For the y-coordinate:
\[ y_O = \frac{1 \cdot (-6) + 6 \cdot 8}{1 + 6} = \frac{-6 + 48}{7} = \frac{42}{7} = 6 \]
Thus, the coordinates of point \( O \) are \( (1, 6) \).
Therefore, the coordinates of point \( O \) are:
\[ \boxed{(1, 6)} \]
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