If the point dividing internally the line segment joining the points (a, b) and (5, 7) in the ratio 2 : 1 be (4, 6), then
(a)a = 1, b = 2
(b) a = 2, b = - 4
(c) a = 2, b = 4
(d)a = - 2, b = 4

Options

A. a = 1, b = 2
B. a = 2, b = - 4
C. a = 2, b = 4
D. a = - 2, b = 4

Answer

To solve the problem, we need to find the values of \( a \) and \( b \) such that the point \( (4, 6) \) divides the line segment joining the points \( (a, b) \) and \( (5, 7) \) in the ratio \( 2:1 \).

### Step 1: Use the section formula
The section formula states that if a point \( (x, y) \) divides the line segment joining the points \( (x_1, y_1) \) and \( (x_2, y_2) \) in the ratio \( m:n \), then:
\[ x = \frac{mx_2 + nx_1}{m+n} \]
\[ y = \frac{my_2 + ny_1}{m+n} \]

### Step 2: Assign values
In our case:
- \( (x_1, y_1) = (a, b) \)
- \( (x_2, y_2) = (5, 7) \)
- The ratio \( m:n = 2:1 \)

### Step 3: Set up the equations
Using the section formula, we can write the equations for \( x \) and \( y \):
\[ 4 = \frac{2 \cdot 5 + 1 \cdot a}{2 + 1} \]
\[ 6 = \frac{2 \cdot 7 + 1 \cdot b}{2 + 1} \]

### Step 4: Simplify the equations
For the \( x \)-coordinate:
\[ 4 = \frac{10 + a}{3} \]
Multiplying both sides by 3:
\[ 12 = 10 + a \]
Subtracting 10 from both sides:
\[ a = 2 \]

For the \( y \)-coordinate:
\[ 6 = \frac{14 + b}{3} \]
Multiplying both sides by 3:
\[ 18 = 14 + b \]
Subtracting 14 from both sides:
\[ b = 4 \]

### Step 5: Conclusion
Thus, we have found:
\[ a = 2, \quad b = 4 \]
The correct option is (c) \( a = 2, b = 4 \).