The point which divides the line segment joining the points (7,-6) and (3,4) in ratio ` 1 : 2` internally lies in the

Options

A. I quadrant
B. II quadrant
C. III quadrant
D. Iv quadrant

Explanation

To find the point that divides the line segment joining the points \( A(7, -6) \) and \( B(3, 4) \) in the ratio \( 1:2 \) internally, we can use the section formula. The section formula states that if a point \( P(x, y) \) divides the line segment joining the points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) in the ratio \( m:n \), then the coordinates of point \( P \) can be calculated using the following formulas:

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

### Step 1: Identify the coordinates and the ratio
- Let \( A(7, -6) \) be \( (x_1, y_1) \) and \( B(3, 4) \) be \( (x_2, y_2) \).
- The ratio \( m:n = 1:2 \) means \( m = 1 \) and \( n = 2 \).

### Step 2: Calculate the x-coordinate of the dividing point
Using the formula for \( x \):

\[ x = \frac{1 \cdot 3 + 2 \cdot 7}{1 + 2} \]
\[ x = \frac{3 + 14}{3} = \frac{17}{3} \]

### Step 3: Calculate the y-coordinate of the dividing point
Using the formula for \( y \):

\[ y = \frac{1 \cdot 4 + 2 \cdot (-6)}{1 + 2} \]
\[ y = \frac{4 - 12}{3} = \frac{-8}{3} \]

### Step 4: Combine the coordinates to find the point
The point \( P \) that divides the line segment in the ratio \( 1:2 \) is:

\[ P\left(\frac{17}{3}, \frac{-8}{3}\right) \]

### Step 5: Determine the quadrant of the point
- The x-coordinate \( \frac{17}{3} \) is positive.
- The y-coordinate \( \frac{-8}{3} \) is negative.

Since the x-coordinate is positive and the y-coordinate is negative, the point lies in the **fourth quadrant**.

### Final Answer
The point which divides the line segment joining the points \( (7, -6) \) and \( (3, 4) \) in the ratio \( 1:2 \) internally is \( \left(\frac{17}{3}, \frac{-8}{3}\right) \) and it lies in the **fourth quadrant**.