What is the ratio in which the point C `(- (2)/(7) - (20)/(7))` divides the line joining the points A (- 2 , -2) and B(2 , -4) ?

Options

A. `1 : 3`
B. `3 : 4`
C. `1: 2`
D. `2 : 3`

Answer

To find the ratio in which the point C divides the line segment joining points A and B, we will use the section formula. The coordinates of the points are given as follows:

- Point A: \( A(-2, -2) \)
- Point B: \( B(2, -4) \)
- Point C: \( C\left(-\frac{2}{7}, -\frac{20}{7}\right) \)

### Step 1: Set up the section formula
The section formula states that if a point \( C(x, y) \) divides the line segment joining points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) in the ratio \( m:n \), then:
\[ x = \frac{mx_2 + nx_1}{m+n} \]
\[ y = \frac{my_2 + ny_1}{m+n} \]

### Step 2: Assign values
Here, we have:
- \( x_1 = -2 \)
- \( y_1 = -2 \)
- \( x_2 = 2 \)
- \( y_2 = -4 \)
- \( x = -\frac{2}{7} \)
- \( y = -\frac{20}{7} \)

### Step 3: Set up equations for x-coordinate
Using the x-coordinate:
\[ -\frac{2}{7} = \frac{m \cdot 2 + n \cdot (-2)}{m+n} \]
This simplifies to:
\[ -\frac{2}{7}(m+n) = 2m - 2n \]
Multiplying through by \( 7 \):
\[ -2(m+n) = 14m - 14n \]
Rearranging gives:
\[ -2m - 2n = 14m - 14n \]
\[ -2m + 14n = 2n \]
\[ 16n = 16m \]
Thus, we have:
\[ \frac{m}{n} = 1 \]

### Step 4: Set up equations for y-coordinate
Now using the y-coordinate:
\[ -\frac{20}{7} = \frac{m \cdot (-4) + n \cdot (-2)}{m+n} \]
This simplifies to:
\[ -\frac{20}{7}(m+n) = -4m - 2n \]
Multiplying through by \( 7 \):
\[ -20(m+n) = -28m - 14n \]
Rearranging gives:
\[ -20m - 20n = -28m - 14n \]
\[ 8m - 6n = 0 \]
Thus, we have:
\[ \frac{m}{n} = \frac{3}{4} \]

### Step 5: Conclusion
From the calculations, we find that the point C divides the line segment AB in the ratio \( 3:4 \).

### Final Answer
The ratio in which point C divides the line joining points A and B is \( 3:4 \).

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