Point P(4,6) divides the line segment `A(-2, 3)` and B(6,7) in the ratio

Options

A. `(2)/(3)`
B. `(1)/(3)`
C. `(3)/(1)`
D. `(3)/(2)`

Answer

To find the ratio in which point P(4,6) divides the line segment joining points A(-2,3) and B(6,7), we can use the section formula. Here’s the step-by-step solution:

### Step 1: Understand 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 M:N, then the coordinates of point P are given by:
\[ P\left(\frac{Mx_2 + Nx_1}{M + N}, \frac{My_2 + Ny_1}{M + N}\right) \]
where (x1, y1) are the coordinates of point A and (x2, y2) are the coordinates of point B.

### Step 2: Assign the Coordinates
Let:
- A = (-2, 3) → (x1, y1)
- B = (6, 7) → (x2, y2)
- P = (4, 6) → (Px, Py)

### Step 3: Set Up the Equation for x-coordinate
Using the section formula for the x-coordinate:
\[ 4 = \frac{M \cdot 6 + N \cdot (-2)}{M + N} \]
Multiply both sides by (M + N):
\[ 4(M + N) = 6M - 2N \]
Expanding gives:
\[ 4M + 4N = 6M - 2N \]

### Step 4: Rearranging the Equation
Rearranging the equation:
\[ 4M + 4N + 2N = 6M \]
\[ 4M + 6N = 6M \]
\[ 6N = 6M - 4M \]
\[ 6N = 2M \]
Dividing both sides by 2:
\[ 3N = M \]

### Step 5: Find the Ratio M:N
From the equation \(3N = M\), we can express the ratio of M to N:
\[ \frac{M}{N} = 3 \implies M:N = 3:1 \]

### Conclusion
Thus, point P(4,6) divides the line segment AB in the ratio of 3:1.

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