A point P divides the line joining the points (2, 1) and (5, -8) in ratio 1:2. Also, the point P lies on the line `2x – y + k = 0`. Find the value of k.

Options

A. 8
B. `-8`
C. `-6`
D. 6

Answer

To solve the problem step by step, we will follow these instructions:

### Step 1: Identify the points and the ratio
We have two points A(2, 1) and B(5, -8). The point P divides the line segment AB in the ratio 1:2.

### Step 2: Use the section formula
The coordinates of point P (x, y) that divides the line segment joining points A(x1, y1) and B(x2, y2) in the ratio m:n can be calculated using the section formula:
\[ P\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right) \]
Here, m = 1, n = 2, \(x_1 = 2\), \(y_1 = 1\), \(x_2 = 5\), and \(y_2 = -8\).

### Step 3: Calculate the x-coordinate of P
Using the formula for the x-coordinate:
\[ x = \frac{1 \cdot 5 + 2 \cdot 2}{1 + 2} = \frac{5 + 4}{3} = \frac{9}{3} = 3 \]

### Step 4: Calculate the y-coordinate of P
Using the formula for the y-coordinate:
\[ y = \frac{1 \cdot (-8) + 2 \cdot 1}{1 + 2} = \frac{-8 + 2}{3} = \frac{-6}{3} = -2 \]

### Step 5: Coordinates of point P
Thus, the coordinates of point P are (3, -2).

### Step 6: Substitute P into the line equation
The point P lies on the line given by the equation \(2x - y + k = 0\). We will substitute the coordinates of P into this equation:
\[ 2(3) - (-2) + k = 0 \]

### Step 7: Simplify the equation
\[ 6 + 2 + k = 0 \\ 8 + k = 0 \]

### Step 8: Solve for k
\[ k = -8 \]

### Conclusion
The value of k is \(-8\).

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