If A and B are the points (-3,4) and (2,1). Then the co -ordinates of pont C on AB produced such that AC=2BC are

Options

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

Answer

To find the coordinates of point C on the line segment AB produced such that AC = 2BC, we can follow these steps:

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

### Step 2: Determine the distance ratios
We know that AC = 2BC. This means that point C divides the line segment AB in the ratio of 2:1.

### Step 3: Apply the section formula
The section formula states that if a point C divides the line segment joining points A(x1, y1) and B(x2, y2) in the ratio m:n, then the coordinates of point C (x, y) can be calculated as:

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

In our case, since AC = 2BC, we have m = 2 and n = 1.

### Step 4: Substitute the values into the section formula
Using the coordinates of A and B:
- \(x_1 = -3\), \(y_1 = 4\)
- \(x_2 = 2\), \(y_2 = 1\)

Now substituting into the formula:

\[ x = \frac{2(2) + 1(-3)}{2 + 1} = \frac{4 - 3}{3} = \frac{1}{3} \]

\[ y = \frac{2(1) + 1(4)}{2 + 1} = \frac{2 + 4}{3} = \frac{6}{3} = 2 \]

### Step 5: Calculate the coordinates of point C
Thus, the coordinates of point C are:

\[ C\left(\frac{1}{3}, 2\right) \]

### Step 6: Verify the calculation
To verify, we can check the distances:
- AC should be twice the length of BC.

### Final Answer:
The coordinates of point C are \(\left(\frac{1}{3}, 2\right)\).

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