The harmonic conjugate of (4,-2) with respect to (2,-4) and (7,1) is

Options

A. (-8,-14)
B. (2,3)
C. (-2,-3)
D. (1,1)

Answer

To find the harmonic conjugate of the point \( P(4, -2) \) with respect to the points \( A(2, -4) \) and \( B(7, 1) \), we will follow these steps:

### Step 1: Identify the points
Let:
- \( A(2, -4) \)
- \( B(7, 1) \)
- \( P(4, -2) \)

### Step 2: Use the section formula to find the ratio in which \( P \) divides \( AB \)
Let the point \( P \) divide the line segment \( AB \) in the ratio \( \lambda : 1 \). According to the section formula, the coordinates of point \( P \) can be expressed as:

\[ P_x = \frac{B_x \cdot \lambda + A_x \cdot 1}{\lambda + 1} \]
\[ P_y = \frac{B_y \cdot \lambda + A_y \cdot 1}{\lambda + 1} \]

Substituting the coordinates of points \( A \) and \( B \):

\[ 4 = \frac{7\lambda + 2}{\lambda + 1} \]
\[ -2 = \frac{1\lambda - 4}{\lambda + 1} \]

### Step 3: Solve the equations to find \( \lambda \)
1. From the first equation:
\[ 4(\lambda + 1) = 7\lambda + 2 \]
\[ 4\lambda + 4 = 7\lambda + 2 \]
\[ 4 - 2 = 7\lambda - 4\lambda \]
\[ 2 = 3\lambda \implies \lambda = \frac{2}{3} \]

2. We can verify with the second equation:
\[ -2(\lambda + 1) = \lambda - 4 \]
\[ -2\lambda - 2 = \lambda - 4 \]
\[ -2 + 4 = \lambda + 2\lambda \]
\[ 2 = 3\lambda \implies \lambda = \frac{2}{3} \]

### Step 4: Find the harmonic conjugate
The harmonic conjugate \( P' \) will divide the same line segment \( AB \) in the ratio \( -\lambda : 1 \) or \( -\frac{2}{3} : 1 \) (externally).

Using the section formula again for the external division:

\[ P'_x = \frac{B_x \cdot (-\lambda) + A_x \cdot 1}{-\lambda + 1} \]
\[ P'_y = \frac{B_y \cdot (-\lambda) + A_y \cdot 1}{-\lambda + 1} \]

Substituting \( \lambda = \frac{2}{3} \):

\[ P'_x = \frac{7 \cdot -\frac{2}{3} + 2 \cdot 1}{-\frac{2}{3} + 1} = \frac{-\frac{14}{3} + 2}{-\frac{2}{3} + 1} = \frac{-\frac{14}{3} + \frac{6}{3}}{\frac{1}{3}} = \frac{-\frac{8}{3}}{\frac{1}{3}} = -8 \]

\[ P'_y = \frac{1 \cdot -\frac{2}{3} + (-4) \cdot 1}{-\frac{2}{3} + 1} = \frac{-\frac{2}{3} - 4}{\frac{1}{3}} = \frac{-\frac{2}{3} - \frac{12}{3}}{\frac{1}{3}} = \frac{-\frac{14}{3}}{\frac{1}{3}} = -14 \]

### Final Answer
Thus, the harmonic conjugate of the point \( (4, -2) \) with respect to the points \( (2, -4) \) and \( (7, 1) \) is:

\[ \boxed{(-8, -14)} \]