Find the ratio in which the line segment joining the points (-1,7)and(4,-3) is divided by the poin(1,3).

Answer

To find the ratio in which the line segment joining the points (-1, 7) and (4, -3) is divided by the point (1, 3), we can use the section formula. Here’s a step-by-step solution:

### Step 1: Identify the points
Let:
- Point A = (-1, 7) which is (x1, y1)
- Point B = (4, -3) which is (x2, y2)
- Point P = (1, 3) which divides the segment AB in the ratio k:1.

### Step 2: Use the section formula
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 k:1, then:
- The x-coordinate of point P is given by:
\[ x = \frac{mx_2 + nx_1}{m+n} \]
- The y-coordinate of point P is given by:
\[ y = \frac{my_2 + ny_1}{m+n} \]

### Step 3: Set up the equations
Using the coordinates of the points:
- For the x-coordinate:
\[ 1 = \frac{4k + (-1)}{k + 1} \]
Simplifying this gives:
\[ 1(k + 1) = 4k - 1 \]
\[ k + 1 = 4k - 1 \]
\[ 2 = 3k \]
\[ k = \frac{2}{3} \]

- For the y-coordinate:
\[ 3 = \frac{-3k + 7}{k + 1} \]
Simplifying this gives:
\[ 3(k + 1) = -3k + 7 \]
\[ 3k + 3 = -3k + 7 \]
\[ 6k = 4 \]
\[ k = \frac{2}{3} \]

### Step 4: Conclusion
Both equations yield the same value of k. Therefore, the ratio in which the line segment joining the points (-1, 7) and (4, -3) is divided by the point (1, 3) is:
\[ \text{Ratio} = k:1 = \frac{2}{3}:1 = 2:3 \]