The point which divides the line segment of points P(-1, 7)and Q(4, -3) in the ratio of `2 : 3` is:

Options

A. (-1, 3)
B. (-1, -3)
C. (1, -3)
D. (1, 3)

Answer

To find the point that divides the line segment joining points P(-1, 7) and Q(4, -3) in the ratio of 2:3, we can use the section formula. The section formula states that if a point R divides the line segment joining points (x1, y1) and (x2, y2) in the ratio m:n, then the coordinates of point R (x, y) can be calculated as follows:

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

### Step 1: Identify the coordinates and the ratio
Given:
- P(-1, 7) → (x1, y1) = (-1, 7)
- Q(4, -3) → (x2, y2) = (4, -3)
- Ratio m:n = 2:3 → m = 2, n = 3

### Step 2: Substitute the values into the formula for x
Using the section formula for x:
\[ x = \frac{mx_2 + nx_1}{m+n} = \frac{2 \cdot 4 + 3 \cdot (-1)}{2 + 3} \]
Calculating the numerator:
\[ = \frac{8 - 3}{5} = \frac{5}{5} = 1 \]

### Step 3: Substitute the values into the formula for y
Using the section formula for y:
\[ y = \frac{my_2 + ny_1}{m+n} = \frac{2 \cdot (-3) + 3 \cdot 7}{2 + 3} \]
Calculating the numerator:
\[ = \frac{-6 + 21}{5} = \frac{15}{5} = 3 \]

### Step 4: Write the coordinates of point R
Thus, the coordinates of point R that divides the line segment PQ in the ratio of 2:3 are:
\[ R(1, 3) \]

### Final Answer
The point which divides the line segment joining P(-1, 7) and Q(4, -3) in the ratio of 2:3 is (1, 3).