If point `C(-4,1)` divides the line segment joining the point `A(2,-2)` and B in the ratio `3:5`, then the coordinates of B are

Options

A. (a) `(-14,6)`
B. (b) `(6,-14)`
C. (c) `(-14,-6)`
D. (d) `(-6,-14)`

Answer

To find the coordinates of point B, we can use the section formula, which states that if a point C divides the line segment joining points A and B in the ratio m:n, then the coordinates of point C can be calculated using the following formulas:

\[ x = \frac{mx_2 + nx_1}{m+n} \]
\[ y = \frac{my_2 + ny_1}{m+n} \]

Where:
- \( (x_1, y_1) \) are the coordinates of point A,
- \( (x_2, y_2) \) are the coordinates of point B,
- \( (x, y) \) are the coordinates of point C,
- \( m \) and \( n \) are the ratios in which point C divides the line segment.

### Step 1: Identify the given points and the ratio
Given:
- Point A \( A(2, -2) \) which means \( x_1 = 2 \) and \( y_1 = -2 \)
- Point C \( C(-4, 1) \) which means \( x = -4 \) and \( y = 1 \)
- The ratio \( m:n = 3:5 \) which means \( m = 3 \) and \( n = 5 \)

### Step 2: Use the section formula for the x-coordinate
Using the section formula for the x-coordinate:
\[ -4 = \frac{3x_2 + 5 \cdot 2}{3 + 5} \]
This simplifies to:
\[ -4 = \frac{3x_2 + 10}{8} \]

### Step 3: Clear the fraction by multiplying both sides by 8
\[ -4 \cdot 8 = 3x_2 + 10 \]
\[ -32 = 3x_2 + 10 \]

### Step 4: Solve for \( x_2 \)
Subtract 10 from both sides:
\[ -32 - 10 = 3x_2 \]
\[ -42 = 3x_2 \]
Now, divide by 3:
\[ x_2 = \frac{-42}{3} = -14 \]

### Step 5: Use the section formula for the y-coordinate
Now, using the section formula for the y-coordinate:
\[ 1 = \frac{3y_2 + 5 \cdot (-2)}{3 + 5} \]
This simplifies to:
\[ 1 = \frac{3y_2 - 10}{8} \]

### Step 6: Clear the fraction by multiplying both sides by 8
\[ 1 \cdot 8 = 3y_2 - 10 \]
\[ 8 = 3y_2 - 10 \]

### Step 7: Solve for \( y_2 \)
Add 10 to both sides:
\[ 8 + 10 = 3y_2 \]
\[ 18 = 3y_2 \]
Now, divide by 3:
\[ y_2 = \frac{18}{3} = 6 \]

### Step 8: Write the coordinates of point B
Thus, the coordinates of point B are:
\[ B(x_2, y_2) = (-14, 6) \]

### Conclusion
The coordinates of point B are \( (-14, 6) \).

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