To find the coordinates of point P that divides the line segment NO in the ratio \( NP:OP = 1:3 \), we can use the section formula.
The coordinates of points N and O are:
- \( N(-9, 1) \)
- \( O(7, 9) \)
Let \( P(x, y) \) be the point dividing the segment in the ratio \( m:n = 1:3 \). According to the section formula, the coordinates of point P can be calculated as follows:
\[ x = \frac{mx_2 + nx_1}{m+n} \] \[ y = \frac{my_2 + ny_1}{m+n} \]
where \( (x_1, y_1) \) are the coordinates of point N, and \( (x_2, y_2) \) are the coordinates of point O.
Substituting the values:
- \( m = 1 \)
- \( n = 3 \)
- \( (x_1, y_1) = (-9, 1) \)
- \( (x_2, y_2) = (7, 9) \)
Calculate the coordinates of P:
For x-coordinate: \[ x = \frac{1 \cdot 7 + 3 \cdot (-9)}{1 + 3} = \frac{7 - 27}{4} = \frac{-20}{4} = -5 \]
For y-coordinate: \[ y = \frac{1 \cdot 9 + 3 \cdot 1}{1 + 3} = \frac{9 + 3}{4} = \frac{12}{4} = 3 \]
Final coordinates of point P:
Thus, the coordinates of point P are: \[ P(-5, 3) \]