What are the coordinates of the point c, such that `B(1/2,6)` divides the line segment joining the points A(3, 5) and C in the ratio of 1 : 3 ?

Options

A. (0, 0)
B. (7, 9)
C. (7, -9)
D. (-7, 9)

Answer

To find the coordinates of point C such that point B(1/2, 6) divides the line segment joining points A(3, 5) and C in the ratio of 1:3, we can use the section formula.

### Step-by-Step Solution:

1. **Identify the points and the ratio**:
- Let A = (3, 5)
- Let B = (1/2, 6)
- Let C = (x, y) (coordinates we need to find)
- The ratio in which B divides AC is 1:3.

2. **Use the section formula**:
The section formula states that if a point B divides the line segment joining points A(x1, y1) and C(x2, y2) in the ratio m:n, then the coordinates of point B are given by:
\[ B\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right) \]
Here, m = 1, n = 3, and we can substitute the coordinates of A and C.

3. **Set up the equations for x-coordinate**:
Using the section formula for the x-coordinate:
\[ \frac{1 \cdot x + 3 \cdot 3}{1 + 3} = \frac{1}{2} \]
Simplifying this gives:
\[ \frac{x + 9}{4} = \frac{1}{2} \]

4. **Multiply both sides by 4**:
\[ x + 9 = 2 \]

5. **Solve for x**:
\[ x = 2 - 9 = -7 \]

6. **Set up the equations for y-coordinate**:
Using the section formula for the y-coordinate:
\[ \frac{1 \cdot y + 3 \cdot 5}{1 + 3} = 6 \]
Simplifying this gives:
\[ \frac{y + 15}{4} = 6 \]

7. **Multiply both sides by 4**:
\[ y + 15 = 24 \]

8. **Solve for y**:
\[ y = 24 - 15 = 9 \]

9. **Final coordinates of point C**:
Therefore, the coordinates of point C are:
\[ C(-7, 9) \]

### Summary:
The coordinates of point C are (-7, 9).