The point which divides the line segment joining the points (7,-6) and (3,4) in ration ` 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 (7, -6) and (3, 4) in the ratio 1:2 internally, we can 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 m1:m2, then the coordinates of point P can be calculated using the following formulas:

\[ P_x = \frac{m1 \cdot x2 + m2 \cdot x1}{m1 + m2} \]
\[ P_y = \frac{m1 \cdot y2 + m2 \cdot y1}{m1 + m2} \]

### Step 1: Identify the coordinates and ratio
Here, we have:
- A(7, -6) → (x1, y1)
- B(3, 4) → (x2, y2)
- The ratio m1:m2 = 1:2, so m1 = 1 and m2 = 2.

### Step 2: Calculate the x-coordinate of point P
Using the section formula for the x-coordinate:
\[ P_x = \frac{1 \cdot 3 + 2 \cdot 7}{1 + 2} \]
Calculating this:
\[ P_x = \frac{3 + 14}{3} = \frac{17}{3} \]

### Step 3: Calculate the y-coordinate of point P
Using the section formula for the y-coordinate:
\[ P_y = \frac{1 \cdot 4 + 2 \cdot (-6)}{1 + 2} \]
Calculating this:
\[ P_y = \frac{4 - 12}{3} = \frac{-8}{3} \]

### Step 4: Combine the coordinates
Thus, the coordinates of point P that divides the line segment in the ratio 1:2 are:
\[ P\left(\frac{17}{3}, \frac{-8}{3}\right) \]

### Step 5: Determine the quadrant
Now, we need to determine in which quadrant the point lies. The x-coordinate is \(\frac{17}{3}\) (positive) and the y-coordinate is \(\frac{-8}{3}\) (negative).

- In the first quadrant, both coordinates are positive.
- In the second quadrant, x is negative and y is positive.
- In the third quadrant, both coordinates are negative.
- In the fourth quadrant, x is positive and y is negative.

Since \(P_x\) is positive and \(P_y\) is negative, point P 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 lies in the **fourth quadrant**.

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