If `A & B` are the points `(-3,4)a n d(2,1)` , then the co-ordinates of the point `C on AB` produced such that `A C=2B C` are: a. (2,4)` b. `(3,7)` c. `(7,-2)` d. `(1/2,5/2)`

Options

A. `(2,4)`
B. (3,7)
C. (7,-2)
D. `(-1/2,5/2)`

Answer

To find the coordinates of point C on line segment AB produced such that AC = 2BC, we can use the section formula. Here are the steps to solve the problem:

### Step 1: Identify the coordinates of points A and B
Given:
- A = (-3, 4)
- B = (2, 1)

### Step 2: Understand the ratio in which point C divides the line segment
We know that AC = 2BC, which means that point C divides the line segment AB externally in the ratio 2:1.

### Step 3: Apply the section formula for external division
The section formula for external division is given by:
\[ x = \frac{m \cdot x_2 - n \cdot x_1}{m - n} \]
\[ y = \frac{m \cdot y_2 - n \cdot y_1}{m - n} \]
where:
- \( (x_1, y_1) \) are the coordinates of point A,
- \( (x_2, y_2) \) are the coordinates of point B,
- m and n are the parts of the ratio.

Here, \( m = 2 \) and \( n = 1 \).

### Step 4: Substitute the values into the section formula
Substituting the values into the formula:
- For x-coordinate:
\[ x = \frac{2 \cdot 2 - 1 \cdot (-3)}{2 - 1} = \frac{4 + 3}{1} = \frac{7}{1} = 7 \]

- For y-coordinate:
\[ y = \frac{2 \cdot 1 - 1 \cdot 4}{2 - 1} = \frac{2 - 4}{1} = \frac{-2}{1} = -2 \]

### Step 5: Write the coordinates of point C
Thus, the coordinates of point C are:
\[ C = (7, -2) \]

### Conclusion
The correct answer is option (c) \( (7, -2) \).

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