The ratio in which the line 3x - 2y + 5 = 0 divides the join of (6,-7) and (-2 , 3) is

Options

A. `1 : 1`
B. `7 : 37`
C. ` 37 : 7 `
D. none of these

Answer

To find the ratio in which the line \(3x - 2y + 5 = 0\) divides the line segment joining the points \(A(6, -7)\) and \(B(-2, 3)\), we can follow these steps:

### Step 1: Find the equation of the line segment AB
The coordinates of points \(A\) and \(B\) are \(A(6, -7)\) and \(B(-2, 3)\). The slope \(m\) of line segment \(AB\) can be calculated using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{3 - (-7)}{-2 - 6} = \frac{10}{-8} = -\frac{5}{4} \]
Using the point-slope form of the equation of a line, we can write the equation of line \(AB\):
\[ y - y_1 = m(x - x_1) \]
Substituting \(m = -\frac{5}{4}\), \(x_1 = 6\), and \(y_1 = -7\):
\[ y + 7 = -\frac{5}{4}(x - 6) \]
Multiplying through by 4 to eliminate the fraction:
\[ 4(y + 7) = -5(x - 6) \]
Expanding and rearranging gives:
\[ 5x + 4y = 2 \]

### Step 2: Find the point of intersection (Q) of the two lines
We have the equations:
1. \(3x - 2y + 5 = 0\) (Equation of the given line)
2. \(5x + 4y = 2\) (Equation of line segment AB)

To find the intersection, we can solve these two equations simultaneously. Rearranging the first equation gives:
\[ 2y = 3x + 5 \implies y = \frac{3}{2}x + \frac{5}{2} \]
Substituting \(y\) in the second equation:
\[ 5x + 4\left(\frac{3}{2}x + \frac{5}{2}\right) = 2 \]
Simplifying:
\[ 5x + 6x + 10 = 2 \implies 11x + 10 = 2 \implies 11x = -8 \implies x = -\frac{8}{11} \]
Now substituting \(x\) back to find \(y\):
\[ y = \frac{3}{2}\left(-\frac{8}{11}\right) + \frac{5}{2} = -\frac{24}{22} + \frac{55}{22} = \frac{31}{22} \]
Thus, the point of intersection \(Q\) is:
\[ Q\left(-\frac{8}{11}, \frac{31}{22}\right) \]

### Step 3: Use the section formula to find the ratio
Let the ratio in which \(Q\) divides \(AB\) be \(m:n\). According to the section formula, the coordinates of point \(Q\) can be expressed as:
\[ Q\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right) \]
Where \(A(6, -7) = (x_1, y_1)\) and \(B(-2, 3) = (x_2, y_2)\).

Setting the x-coordinates:
\[ -\frac{8}{11} = \frac{m(-2) + n(6)}{m+n} \]
Cross-multiplying gives:
\[ -8(m+n) = -2m + 6n \implies -8m - 8n = -2m + 6n \implies 6n + 8n = 8m \implies 14n = 14m \implies m = n \]

Now for the y-coordinates:
\[ \frac{31}{22} = \frac{m(3) + n(-7)}{m+n} \]
Cross-multiplying gives:
\[ 31(m+n) = 3m - 7n \implies 31m + 31n = 3m - 7n \implies 31m + 38n = 0 \]
This implies:
\[ m = -\frac{38}{31}n \]

### Step 4: Finding the ratio
From the equations derived, we can find the ratio \(m:n\):
\[ m:n = 38:31 \]

### Final Answer
The ratio in which the line \(3x - 2y + 5 = 0\) divides the line segment joining points \(A(6, -7)\) and \(B(-2, 3)\) is \(37:7\).