Find the ratio in which the points
Q(-1,-8) divide the line segment joining the points A(1,-2) and B(4,7).

Answer

To find the ratio in which the point Q(-1, -8) divides the line segment joining the points A(1, -2) and B(4, 7), we can use the section formula. The section formula states that if a point divides a line segment joining two points A(x1, y1) and B(x2, y2) in the ratio m:n, then the coordinates of the point dividing the segment can be given by:

\[ \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) \]

### Step 1: Assign the coordinates
Let A(1, -2) and B(4, 7). Let the ratio in which Q divides AB be k:1.

### Step 2: Apply the section formula
Using the section formula, the coordinates of point Q can be expressed as:

\[ Q = \left( \frac{k \cdot 4 + 1 \cdot 1}{k + 1}, \frac{k \cdot 7 + 1 \cdot (-2)}{k + 1} \right) \]

### Step 3: Set up equations for x and y coordinates
Since Q is given as (-1, -8), we can set up two equations based on the x and y coordinates:

1. For the x-coordinate:
\[ \frac{4k + 1}{k + 1} = -1 \]

2. For the y-coordinate:
\[ \frac{7k - 2}{k + 1} = -8 \]

### Step 4: Solve the first equation
Starting with the x-coordinate equation:

\[ 4k + 1 = -1(k + 1) \]
\[ 4k + 1 = -k - 1 \]
\[ 4k + k = -1 - 1 \]
\[ 5k = -2 \]
\[ k = -\frac{2}{5} \]

### Step 5: Solve the second equation (for verification)
Now, let's solve the y-coordinate equation:

\[ 7k - 2 = -8(k + 1) \]
\[ 7k - 2 = -8k - 8 \]
\[ 7k + 8k = -8 + 2 \]
\[ 15k = -6 \]
\[ k = -\frac{2}{5} \]

### Step 6: Interpret the result
The value of k is -2/5, which indicates that the point Q divides the line segment externally. The ratio of the segments QA to BQ is given by the absolute value of k, which is 2:5.

### Final Answer
Thus, the ratio in which the point Q(-1, -8) divides the line segment AB is 2:5.

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