In what ratio does `(4,1)` divide the line segment joining the point `(1,-2)` and `(5,2)`.

Answer

To find the ratio in which the point \( (4, 1) \) divides the line segment joining the points \( (1, -2) \) and \( (5, 2) \), 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:

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

### Step 1: Assign the points
Let:
- \( A(1, -2) \) be \( (x_1, y_1) \)
- \( B(5, 2) \) be \( (x_2, y_2) \)
- \( P(4, 1) \) be the point that divides the line segment.

### Step 2: Set up the equations
Assume the ratio in which \( P \) divides \( AB \) is \( m:n \). Then, we can write the equations for \( x \) and \( y \):
\[ 4 = \frac{m \cdot 5 + n \cdot 1}{m+n} \]
\[ 1 = \frac{m \cdot 2 + n \cdot (-2)}{m+n} \]

### Step 3: Solve the first equation for \( m \) and \( n \)
From the first equation:
\[ 4(m+n) = 5m + n \]
Expanding and rearranging gives:
\[ 4m + 4n = 5m + n \implies 4n - n = 5m - 4m \implies 3n = m \implies m = 3n \]

### Step 4: Substitute \( m \) in the second equation
Now substitute \( m = 3n \) into the second equation:
\[ 1 = \frac{(3n) \cdot 2 + n \cdot (-2)}{3n + n} \]
This simplifies to:
\[ 1 = \frac{6n - 2n}{4n} = \frac{4n}{4n} = 1 \]
This equation is always true, confirming our ratio.

### Step 5: Find the ratio
Since \( m = 3n \), we can express the ratio \( m:n \) as:
\[ m:n = 3n:n = 3:1 \]

### Final Answer
Thus, the point \( (4, 1) \) divides the line segment joining \( (1, -2) \) and \( (5, 2) \) in the ratio \( 3:1 \).

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