To find the coordinates of point Q that divides the line segment OP in the ratio 1:3, we can use the section formula. The section formula states that if a point divides a line segment joining two points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) in the ratio \( m:n \), the coordinates of the point \( P(x, y) \) can be calculated as:
\[ x = \frac{mx_2 + nx_1}{m+n} \] \[ y = \frac{my_2 + ny_1}{m+n} \]
Here, point O has coordinates \( O(-12, 8) \) and point P has coordinates \( P(0, -4) \), and the ratio \( OQ:PQ = 1:3 \). Thus, we have \( m = 1 \) and \( n = 3 \).
Now we can substitute into the formulas:
- Calculating the x-coordinate of Q:
\[ x_Q = \frac{1 \cdot 0 + 3 \cdot (-12)}{1 + 3} = \frac{0 - 36}{4} = \frac{-36}{4} = -9 \]
- Calculating the y-coordinate of Q:
\[ y_Q = \frac{1 \cdot (-4) + 3 \cdot 8}{1 + 3} = \frac{-4 + 24}{4} = \frac{20}{4} = 5 \]
Therefore, the coordinates of point Q are:
\[ Q(-9, 5) \]
Thus, the final answer is:
\[ \text{Coordinates of Q: } (-9, 5) \]
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