Given points `A(4,5),B(-1,-4),C(1,3),D(5,-3)`,then the ratio of the segments into which AB is divided by CD is

Options

A. `13:20`
B. `11:19`
C. `21:29`
D. None of these

Answer

To find the ratio of the segments into which line segment AB is divided by line segment CD, we will follow these steps:

### Step 1: Find the equations of lines AB and CD

**For line AB:**
Given points A(4, 5) and B(-1, -4), we can use the two-point form of the line equation:

\[ y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1) \]

Substituting the coordinates of points A and B:

\[ y - 5 = \frac{-4 - 5}{-1 - 4}(x - 4) \]

Calculating the slope:

\[ y - 5 = \frac{-9}{-5}(x - 4) \]

This simplifies to:

\[ y - 5 = \frac{9}{5}(x - 4) \]

Multiplying through by 5 to eliminate the fraction:

\[ 5y - 25 = 9x - 36 \]

Rearranging gives us the equation of line AB:

\[ 9x - 5y - 11 = 0 \quad \text{(Equation 1)} \]

**For line CD:**
Given points C(1, 3) and D(5, -3), we use the same formula:

\[ y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1) \]

Substituting the coordinates of points C and D:

\[ y - 3 = \frac{-3 - 3}{5 - 1}(x - 1) \]

Calculating the slope:

\[ y - 3 = \frac{-6}{4}(x - 1) \]

This simplifies to:

\[ y - 3 = -\frac{3}{2}(x - 1) \]

Multiplying through by 2:

\[ 2y - 6 = -3(x - 1) \]

Rearranging gives us the equation of line CD:

\[ 3x + 2y - 9 = 0 \quad \text{(Equation 2)} \]

### Step 2: Find the intersection point of lines AB and CD

To find the intersection point, we solve the two equations simultaneously:

1. \(9x - 5y - 11 = 0\)
2. \(3x + 2y - 9 = 0\)

From Equation 2, we can express y in terms of x:

\[ 2y = 9 - 3x \implies y = \frac{9 - 3x}{2} \]

Substituting this expression for y into Equation 1:

\[ 9x - 5\left(\frac{9 - 3x}{2}\right) - 11 = 0 \]

Multiplying through by 2 to eliminate the fraction:

\[ 18x - 5(9 - 3x) - 22 = 0 \]

Expanding:

\[ 18x - 45 + 15x - 22 = 0 \]

Combining like terms:

\[ 33x - 67 = 0 \implies x = \frac{67}{33} \]

Now substituting x back to find y:

\[ y = \frac{9 - 3\left(\frac{67}{33}\right)}{2} = \frac{9 - \frac{201}{33}}{2} = \frac{\frac{297 - 201}{33}}{2} = \frac{\frac{96}{33}}{2} = \frac{48}{33} \]

Thus, the intersection point R is:

\[ R\left(\frac{67}{33}, \frac{48}{33}\right) \]

### Step 3: Use the section formula to find the ratio

Using the section formula, if point R divides line segment AB in the ratio \(m:n\), then:

\[ \frac{mx_2 + nx_1}{m+n} = x_R \quad \text{and} \quad \frac{my_2 + ny_1}{m+n} = y_R \]

Substituting the coordinates of A(4, 5) and B(-1, -4):

For x-coordinate:

\[ \frac{m(-1) + n(4)}{m+n} = \frac{67}{33} \]

For y-coordinate:

\[ \frac{m(-4) + n(5)}{m+n} = \frac{48}{33} \]

### Step 4: Solve for m and n

From the x-coordinate equation:

\[ m(-1) + n(4) = \frac{67(m+n)}{33} \]

From the y-coordinate equation:

\[ m(-4) + n(5) = \frac{48(m+n)}{33} \]

Solving these equations will yield the values of m and n, which will give us the ratio \(m:n\).

After solving, we find:

\[ m:n = 13:20 \]

### Final Answer:
The ratio of the segments into which AB is divided by CD is **13:20**.