In what ratio will the point `((1)/(2),(-13)/(4))` internally divide the line segment joining the point (3,-5) and(-7, 2)?

Options

A. `(1)/(3)`
B. `(1)/(4)`
C. `(2)/(3)`
D. `(1)/(5)`

Answer

To find the ratio in which the point \(\left(\frac{1}{2}, -\frac{13}{4}\right)\) divides the line segment joining the points \((3, -5)\) and \((-7, 2)\), we will use the section formula.

### Step-by-Step Solution:

1. **Identify the points and the coordinates of the dividing point:**
- Let \(A(3, -5)\) and \(B(-7, 2)\) be the endpoints of the line segment.
- Let \(P\left(\frac{1}{2}, -\frac{13}{4}\right)\) be the point that divides the segment.

2. **Assume the ratio in which point \(P\) divides the segment:**
- Let the ratio be \(m:n\).

3. **Use the section formula:**
- The coordinates of point \(P\) can be expressed using the section formula:
\[ P\left(x, y\right) = \left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right) \]
- Here, \(x_1 = 3\), \(y_1 = -5\), \(x_2 = -7\), and \(y_2 = 2\).

4. **Set up the equations for \(x\) and \(y\):**
- For the x-coordinate:
\[ \frac{m(-7) + n(3)}{m+n} = \frac{1}{2} \]
- For the y-coordinate:
\[ \frac{m(2) + n(-5)}{m+n} = -\frac{13}{4} \]

5. **Cross-multiply to eliminate the denominators:**
- For the x-coordinate:
\[ 2(m(-7) + n(3)) = (m+n) \]
\[ -14m + 6n = m + n \]
\[ -15m + 5n = 0 \quad \text{(Equation 1)} \]
- For the y-coordinate:
\[ 4(m(2) + n(-5)) = -13(m+n) \]
\[ 8m - 20n = -13m - 13n \]
\[ 21m - 7n = 0 \quad \text{(Equation 2)} \]

6. **Solve the equations:**
- From Equation 1:
\[ 15m = 5n \implies \frac{m}{n} = \frac{1}{3} \]
- From Equation 2:
\[ 21m = 7n \implies \frac{m}{n} = \frac{1}{3} \]

7. **Conclusion:**
- The point \(\left(\frac{1}{2}, -\frac{13}{4}\right)\) divides the line segment joining the points \((3, -5)\) and \((-7, 2)\) in the ratio \(1:3\).