The equation of the line which passes through (4,7) and divides the join of (1,7) and (6,-3) internally in the ratio 2:3, is

Options

A. y=4x-9
B. x=4y-9
C. 4x+y=9
D. none of these

Answer

To find the equation of the line that passes through the point (4, 7) and divides the line segment joining the points (1, 7) and (6, -3) internally in the ratio 2:3, we can follow these steps:

### Step 1: Find the coordinates of the point that divides the line segment
We will use the section formula to find the coordinates of the point that divides the line segment joining (1, 7) and (6, -3) in the ratio 2:3.

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 (x, y) can be calculated as:

\[ x = \frac{m_1 x_2 + m_2 x_1}{m_1 + m_2} \]
\[ y = \frac{m_1 y_2 + m_2 y_1}{m_1 + m_2} \]

Here, A(1, 7) and B(6, -3) with m1 = 2 and m2 = 3.

Calculating the x-coordinate:
\[ x = \frac{2 \cdot 6 + 3 \cdot 1}{2 + 3} = \frac{12 + 3}{5} = \frac{15}{5} = 3 \]

Calculating the y-coordinate:
\[ y = \frac{2 \cdot (-3) + 3 \cdot 7}{2 + 3} = \frac{-6 + 21}{5} = \frac{15}{5} = 3 \]

Thus, the coordinates of the point that divides the segment are (3, 3).

### Step 2: Use the two-point form of the equation of a line
Now we need to find the equation of the line that passes through the points (4, 7) and (3, 3).

The two-point form of the equation of a line is given by:
\[ y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1) \]

Using (x1, y1) = (4, 7) and (x2, y2) = (3, 3):
\[ y - 7 = \frac{3 - 7}{3 - 4}(x - 4) \]
\[ y - 7 = \frac{-4}{-1}(x - 4) \]
\[ y - 7 = 4(x - 4) \]

### Step 3: Simplify the equation
Now, we simplify the equation:
\[ y - 7 = 4x - 16 \]
\[ y = 4x - 16 + 7 \]
\[ y = 4x - 9 \]

### Step 4: Write in standard form
To write this in standard form (Ax + By + C = 0):
\[ 4x - y - 9 = 0 \]

Thus, the required equation of the line is:
\[ 4x - y - 9 = 0 \]