Find the coordinates of a point which cuts the line segment connecting (4,-3) and (8, 5) internally in ratio 3:1

Question

Accepted Answer

To find the coordinates of a point that divides the line segment joining the points (4, -3) and (8, 5) internally in the ratio 3:1, we use the section formula. The section formula for a point dividing a line segment in the ratio $ m:n $ is given by:

$$
(x, y) = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right)
$$

Here, $ (x_1, y_1) = (4, -3) $ and $ (x_2, y_2) = (8, 5) $, with the ratio $ m:n = 3:1 $.

Substituting these values into the formula, we get:

$$
x = \frac{3 \times 8 + 1 \times 4}{3 + 1} = \frac{24 + 4}{4} = \frac{28}{4} = 7
$$

$$
y = \frac{3 \times 5 + 1 \times (-3)}{3 + 1} = \frac{15 - 3}{4} = \frac{12}{4} = 3
$$

Thus, the coordinates of the point are $ (7, 3) $.

**Explanation for Option 3 being correct:**

- **Option 3: (7, 3)** is indeed the correct answer as calculated above. The point (7, 3) divides the line segment joining (4, -3) and (8, 5) in the ratio 3:1 internally, as per the section formula.

**Explanation for why other options are incorrect:**

- **Option 1: (3, 7)** is incorrect because when applying the section formula, the x-coordinate should be 7 and the y-coordinate should be 3, not the other way around.

- **Option 2: (3, 4)** is incorrect because neither the x-coordinate nor the y-coordinate matches the calculated values of 7 and 3, respectively.

- **Option 4: (3, 3)** is incorrect because while the y-coordinate is 3, the x-coordinate should be 7, not 3.

In conclusion, Option 3 is the only choice that correctly represents the coordinates of the point dividing the line segment in the specified ratio.