A is a point on the positive x-axis at a distance 3 units from the origin and B is a point on the positive y-axis at a distance 4 units from the origin. If P divides AB in the ratio 1 : 2, the coordinates of P are

Options

A. (1, 8/3)
B. (2, 4/3)
C. (8/3, 1)
D. (4/3, 2)

Answer

To find the coordinates of point P that divides the line segment AB in the ratio 1:2, we will follow these steps:

### Step 1: Identify the coordinates of points A and B
- Point A is on the positive x-axis at a distance of 3 units from the origin. Therefore, the coordinates of A are (3, 0).
- Point B is on the positive y-axis at a distance of 4 units from the origin. Therefore, the coordinates of B are (0, 4).

### Step 2: Use the section formula to find the coordinates of point P
The section formula states that if a point P divides the line segment joining points A(x1, y1) and B(x2, y2) in the ratio m:n, then the coordinates of point P (x, y) can be calculated as follows:

\[ x = \frac{m \cdot x_2 + n \cdot x_1}{m+n} \]
\[ y = \frac{m \cdot y_2 + n \cdot y_1}{m+n} \]

In our case:
- A = (3, 0) → (x1, y1) = (3, 0)
- B = (0, 4) → (x2, y2) = (0, 4)
- The ratio m:n = 1:2, so m = 1 and n = 2.

### Step 3: Calculate the x-coordinate of point P
Using the formula for x:

\[ x = \frac{1 \cdot 0 + 2 \cdot 3}{1 + 2} = \frac{0 + 6}{3} = \frac{6}{3} = 2 \]

### Step 4: Calculate the y-coordinate of point P
Using the formula for y:

\[ y = \frac{1 \cdot 4 + 2 \cdot 0}{1 + 2} = \frac{4 + 0}{3} = \frac{4}{3} \]

### Step 5: Combine the results
The coordinates of point P are (2, \(\frac{4}{3}\)).

### Final Answer
The coordinates of point P are \((2, \frac{4}{3})\).

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