To find the equations of the straight lines that divide the line segment joining the points \((-3, 7)\) and \((5, -4)\) in the ratio \(4:7\) both internally and externally, and which are also perpendicular to this line segment, we can follow these steps:
### Step 1: Find the coordinates of the points of division
**Internal Division:**
Using the section formula for internal division, the coordinates of point \(P\) that divides the line segment in the ratio \(m:n = 4:7\) are given by:
\[
P\left(\frac{m x_2 + n x_1}{m+n}, \frac{m y_2 + n y_1}{m+n}\right)
\]
Substituting \(A(-3, 7)\) and \(B(5, -4)\):
\[
P\left(\frac{4 \cdot 5 + 7 \cdot (-3)}{4 + 7}, \frac{4 \cdot (-4) + 7 \cdot 7}{4 + 7}\right)
\]
Calculating the x-coordinate:
\[
x_P = \frac{20 - 21}{11} = \frac{-1}{11}
\]
Calculating the y-coordinate:
\[
y_P = \frac{-16 + 49}{11} = \frac{33}{11} = 3
\]
Thus, the coordinates of point \(P\) are \((-1, 3)\).
**External Division:**
Using the section formula for external division, the coordinates of point \(Q\) that divides the line segment in the ratio \(4:7\) externally are given by:
\[
Q\left(\frac{m x_2 - n x_1}{m-n}, \frac{m y_2 - n y_1}{m-n}\right)
\]
Substituting \(A(-3, 7)\) and \(B(5, -4)\):
\[
Q\left(\frac{4 \cdot 5 - 7 \cdot (-3)}{4 - 7}, \frac{4 \cdot (-4) - 7 \cdot 7}{4 - 7}\right)
\]
Calculating the x-coordinate:
\[
x_Q = \frac{20 + 21}{-3} = \frac{41}{-3} = -\frac{41}{3}
\]
Calculating the y-coordinate:
\[
y_Q = \frac{-16 - 49}{-3} = \frac{-65}{-3} = \frac{65}{3}
\]
Thus, the coordinates of point \(Q\) are \(\left(-\frac{41}{3}, \frac{65}{3}\right)\).
### Step 2: Find the slope of line segment AB
The slope \(m_{AB}\) of the line segment joining points \(A\) and \(B\) is calculated as follows:
\[
m_{AB} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-4 - 7}{5 - (-3)} = \frac{-11}{8}
\]
### Step 3: Find the slope of the perpendicular lines
The slope \(m\) of the lines that are perpendicular to \(AB\) is given by:
\[
m = -\frac{1}{m_{AB}} = -\frac{1}{-\frac{11}{8}} = \frac{8}{11}
\]
### Step 4: Write the equations of the lines
**Equation of the line through point \(P(-1, 3)\):**
Using the point-slope form of the line equation:
\[
y - y_1 = m(x - x_1)
\]
Substituting \(m = \frac{8}{11}\) and point \(P(-1, 3)\):
\[
y - 3 = \frac{8}{11}(x + 1)
\]
Multiplying through by 11 to eliminate the fraction:
\[
11(y - 3) = 8(x + 1)
\]
Expanding and rearranging gives:
\[
11y - 33 = 8x + 8 \implies 8x - 11y + 41 = 0
\]
**Equation of the line through point \(Q\left(-\frac{41}{3}, \frac{65}{3}\right)\):**
Using the same point-slope form:
\[
y - \frac{65}{3} = \frac{8}{11}\left(x + \frac{41}{3}\right)
\]
Multiplying through by 33 to eliminate the fractions:
\[
33\left(y - \frac{65}{3}\right) = 33\cdot\frac{8}{11}\left(x + \frac{41}{3}\right)
\]
Expanding gives:
\[
33y - 715 = 24(x + \frac{41}{3})
\]
Solving this gives:
\[
33y - 715 = 24x + 328 \implies 24x - 33y + 1043 = 0
\]
### Final Equations
1. The equation of the line that divides internally is:
\[
8x - 11y + 41 = 0
\]
2. The equation of the line that divides externally is:
\[
24x - 33y + 1043 = 0
\]
