To solve this problem, we are given that point Q lies on the line segment $OP$, and the ratio $OQ:PQ = 3:2$. We need to determine the coordinates of point Q.

Step-by-Step Solution:

1. Identify the coordinates of O and P:

   • $O = (4, 2)$ (as seen from the graph).
   • $P = (6, -8)$.

2. Apply the section formula: Given that the ratio $OQ : PQ = 3:2$, the section formula gives the coordinates of the point Q dividing the segment $OP$ in the ratio $m:n$: $Q(x, y) = \left( \frac{m \cdot x_2 + n \cdot x_1}{m+n}, \frac{m \cdot y_2 + n \cdot y_1}{m+n} \right)$ where $(x_1, y_1)$ are the coordinates of $O$, and $(x_2, y_2)$ are the coordinates of $P$, and $m:n = 3:2$.

3. Substitute the values:

   • $(x_1, y_1) = (4, 2)$, $(x_2, y_2) = (6, -8)$, and $m = 3$, $n = 2$.

   For the x-coordinate of Q: $x_Q = \frac{3 \cdot 6 + 2 \cdot 4}{3 + 2} = \frac{18 + 8}{5} = \frac{26}{5} = 5.2$

   For the y-coordinate of Q: $y_Q = \frac{3 \cdot (-8) + 2 \cdot 2}{3 + 2} = \frac{-24 + 4}{5} = \frac{-20}{5} = -4$

4. Conclusion: The coordinates of point Q are $Q(5.2, -4)$.

You can now plot this point on the graph by locating $Q(5.2, -4)$.

Do you want further details or have any questions?

Related Questions:

1. How is the section formula derived?
2. What is the difference between internal and external division in coordinate geometry?
3. How would the solution change if the ratio were reversed (i.e., $OQ:PQ = 2:3$)?
4. Can the section formula be used in three-dimensional geometry?
5. How can the distance formula help verify the division of the segment?

Tip:

When using the section formula, always ensure the correct ratio $m:n$ is used and verify the direction of division (internal or external).