To find the harmonic conjugate of the point (4, -2) with respect to the points (2, -4) and (7, 1), we can follow these steps:
### Step 1: Understand the concept of harmonic conjugates
Harmonic conjugates are points that maintain a specific ratio along a line segment. In this case, we want to find the harmonic conjugate of point A (4, -2) with respect to points B (2, -4) and C (7, 1).
### Step 2: Set up the ratio
Let the harmonic conjugate point be D(x, y). The points B and C divide the line segment AD in the ratio k:1. We can express the coordinates of D using the section formula.
### Step 3: Use the section formula
The coordinates of point D can be expressed as:
\[
D\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right)
\]
where \( (x_1, y_1) = (4, -2) \), \( (x_2, y_2) = (2, -4) \), and \( (x_3, y_3) = (7, 1) \). Here, \( m = k \) and \( n = 1 \).
### Step 4: Set up the equations
Using the section formula for both x and y coordinates, we have:
1. For the x-coordinate:
\[
x = \frac{7k + 2}{k + 1}
\]
2. For the y-coordinate:
\[
y = \frac{4k + (-4)}{k + 1}
\]
### Step 5: Set the coordinates equal to the known point
Since D is the harmonic conjugate of A, we can set the x and y coordinates equal to the coordinates of A:
1. For x:
\[
\frac{7k + 2}{k + 1} = 4
\]
2. For y:
\[
\frac{4k - 4}{k + 1} = -2
\]
### Step 6: Solve the equations
**For the x-coordinate equation:**
\[
7k + 2 = 4(k + 1)
\]
\[
7k + 2 = 4k + 4
\]
\[
3k = 2 \quad \Rightarrow \quad k = \frac{2}{3}
\]
**For the y-coordinate equation:**
\[
4k - 4 = -2(k + 1)
\]
\[
4k - 4 = -2k - 2
\]
\[
6k = 2 \quad \Rightarrow \quad k = \frac{1}{3}
\]
### Step 7: Substitute k back into the section formula
Now substitute \( k = \frac{2}{3} \) into the section formula to find the coordinates of D:
1. For x:
\[
x = \frac{7 \cdot \frac{2}{3} + 2}{\frac{2}{3} + 1} = \frac{\frac{14}{3} + 2}{\frac{5}{3}} = \frac{\frac{14}{3} + \frac{6}{3}}{\frac{5}{3}} = \frac{\frac{20}{3}}{\frac{5}{3}} = 4
\]
2. For y:
\[
y = \frac{4 \cdot \frac{2}{3} - 4}{\frac{2}{3} + 1} = \frac{\frac{8}{3} - 4}{\frac{5}{3}} = \frac{\frac{8}{3} - \frac{12}{3}}{\frac{5}{3}} = \frac{-\frac{4}{3}}{\frac{5}{3}} = -\frac{4}{5}
\]
### Final Result
Thus, the harmonic conjugate of (4, -2) with respect to (2, -4) and (7, 1) is:
\[
D\left(4, -\frac{4}{5}\right)
\]
