Find the ratio in which `C(1,-3)` divides the line joining `W(-4,-3)` and `E(5,-3)`.

Options

A. 0.21111111111111
B. 0.21041666666667
C. 0.086805555555556
D. 0.042361111111111

Answer

To find the ratio in which the point C(1, -3) divides the line segment joining the points W(-4, -3) and E(5, -3), we can use the section formula. The section formula states that if a point divides a line segment joining two points in the ratio m:n, then the coordinates of the point can be calculated as follows:

\[ \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) \]

Where (x1, y1) and (x2, y2) are the coordinates of points W and E respectively.

### Step 1: Assign the coordinates
Let:
- W = (x1, y1) = (-4, -3)
- E = (x2, y2) = (5, -3)
- C = (x, y) = (1, -3)

### Step 2: Set up the ratio
Let the ratio in which C divides the line segment WE be λ:1. Thus, we can express the coordinates of C using the section formula:

\[ C = \left( \frac{-4\lambda + 5}{\lambda + 1}, \frac{-3\lambda - 3}{\lambda + 1} \right) \]

### Step 3: Equate the x-coordinates
Since C is at (1, -3), we can set up the equation for the x-coordinates:

\[ 1 = \frac{-4\lambda + 5}{\lambda + 1} \]

### Step 4: Cross-multiply to solve for λ
Cross-multiplying gives:

\[ 1(\lambda + 1) = -4\lambda + 5 \]

This simplifies to:

\[ \lambda + 1 = -4\lambda + 5 \]

### Step 5: Rearrange the equation
Rearranging the equation gives:

\[ \lambda + 4\lambda = 5 - 1 \]
\[ 5\lambda = 4 \]

### Step 6: Solve for λ
Dividing both sides by 5:

\[ \lambda = \frac{4}{5} \]

### Step 7: Write the ratio
Thus, the ratio in which C divides the line segment WE is:

\[ \lambda : 1 = \frac{4}{5} : 1 = 4 : 5 \]

### Step 8: Determine the ratio in terms of W and E
Since we want the ratio in which C divides the line segment from W to E, we can express this as:

\[ C \text{ divides } WE \text{ in the ratio } 5 : 4 \]

### Final Answer:
C divides the line segment joining W and E in the ratio 5:4.