To find the coordinates of the point $P(x, y)$ that divides the line segment joining two points $A(5, 3)$ and $B(1, 3)$ in a given ratio, we can use the section formula.

Internal Division:

For internal division, the formula for the coordinates of the point $P(x, y)$ that divides the line segment joining two points $A(x_1, y_1)$ and $B(x_2, y_2)$ in the ratio $m:n$ is: $P(x, y) = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right)$ Given:

• $A(5, 3)$
• $B(1, 3)$
• Ratio $m:n = 3:1$

Substitute the values: $x = \frac{(3 \cdot 1) + (1 \cdot 5)}{3 + 1} = \frac{3 + 5}{4} = \frac{8}{4} = 2$ $y = \frac{(3 \cdot 3) + (1 \cdot 3)}{3 + 1} = \frac{9 + 3}{4} = \frac{12}{4} = 3$

So, the point $P$ dividing the line segment internally in the ratio 3:1 is $(2, 3)$.

External Division:

For external division, the formula for the coordinates of the point $P(x, y)$ that divides the line segment joining two points $A(x_1, y_1)$ and $B(x_2, y_2)$ externally in the ratio $m:n$ is: $P(x, y) = \left( \frac{mx_2 - nx_1}{m-n}, \frac{my_2 - ny_1}{m-n} \right)$

Substitute the values: $x = \frac{(3 \cdot 1) - (1 \cdot 5)}{3 - 1} = \frac{3 - 5}{2} = \frac{-2}{2} = -1$ $y = \frac{(3 \cdot 3) - (1 \cdot 3)}{3 - 1} = \frac{9 - 3}{2} = \frac{6}{2} = 3$

So, the point $P$ dividing the line segment externally in the ratio 3:1 is $(-1, 3)$.

Final Answer:

• The point dividing the line segment internally in the ratio 3:1 is $(2, 3)$.
• The point dividing the line segment externally in the ratio 3:1 is $(-1, 3)$.

Would you like more details on any part of this? Here are some related questions for further exploration:

1. How is the section formula derived?
2. Can you explain how the section formula changes for different types of divisions?
3. What if the points A and B had different y-coordinates instead of both being 3?
4. How can you apply the section formula in three-dimensional geometry?
5. How is the external division formula used in vector geometry?

Tip: When using the section formula, always ensure the ratio values are consistent (internal or external) and simplify the calculations carefully.