A line perpendicular to the line segment joining the points (7, 3) and (3, 7) divides it in the ratio 1 : 3, the equation of the line is

Options

A. `x+y-10=0`
B. `x-y+4=0`
C. `x-y+2=0`
D. `x-y-2=0`

Answer

To find the equation of the line that is perpendicular to the line segment joining the points (7, 3) and (3, 7) and divides it in the ratio 1:3, we can follow these steps:

### Step 1: Find the coordinates of the point that divides the line segment in the ratio 1:3.
Let the points be \( A(7, 3) \) and \( B(3, 7) \). We can use the section formula to find the coordinates of point \( C \) that divides \( AB \) in the ratio \( m_1:m_2 = 1:3 \).

Using the section formula:
\[ C\left(\frac{m_2 x_1 + m_1 x_2}{m_1 + m_2}, \frac{m_2 y_1 + m_1 y_2}{m_1 + m_2}\right) \]
where \( (x_1, y_1) = (7, 3) \) and \( (x_2, y_2) = (3, 7) \).

Substituting the values:
\[ C\left(\frac{3 \cdot 7 + 1 \cdot 3}{1 + 3}, \frac{3 \cdot 3 + 1 \cdot 7}{1 + 3}\right) \]
Calculating the x-coordinate:
\[ C_x = \frac{21 + 3}{4} = \frac{24}{4} = 6 \]
Calculating the y-coordinate:
\[ C_y = \frac{9 + 7}{4} = \frac{16}{4} = 4 \]
Thus, the coordinates of point \( C \) are \( (6, 4) \).

### Step 2: Find the slope of line segment \( AB \).
The slope \( m \) of line segment \( AB \) can be calculated using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting the coordinates of points \( A \) and \( B \):
\[ m_{AB} = \frac{7 - 3}{3 - 7} = \frac{4}{-4} = -1 \]

### Step 3: Find the slope of the perpendicular line.
If two lines are perpendicular, the product of their slopes is \( -1 \). Therefore, if the slope of \( AB \) is \( -1 \), the slope \( m_2 \) of the perpendicular line is:
\[ m_2 = -\frac{1}{m_{AB}} = -\frac{1}{-1} = 1 \]

### Step 4: Write the equation of the line using point-slope form.
The point-slope form of the equation of a line is given by:
\[ y - y_1 = m(x - x_1) \]
Using point \( C(6, 4) \) and slope \( m_2 = 1 \):
\[ y - 4 = 1(x - 6) \]
Simplifying this:
\[ y - 4 = x - 6 \]
\[ y = x - 2 \]

### Final Equation
The equation of the line is:
\[ x - y - 2 = 0 \]