In the ellipse ` (x^(2))/( a^(2)) + (y^(2))/(b^(2)) = 1 ` write down the equations to the diameters which are conjugate to the diameters whose equations are x - y = 0, x + y = 0, ` y = (a)/(b) x, and y = (b)/(a) x`

Answer

To find the equations of the diameters that are conjugate to the given diameters of the ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), we can follow these steps:

### Step 1: Identify the slopes of the given diameters
The given diameters are:
1. \( x - y = 0 \) (slope \( m_1 = 1 \))
2. \( x + y = 0 \) (slope \( m_2 = -1 \))
3. \( y = \frac{a}{b} x \) (slope \( m_3 = \frac{a}{b} \))
4. \( y = \frac{b}{a} x \) (slope \( m_4 = \frac{b}{a} \))

### Step 2: Use the formula for conjugate diameters
For a given slope \( m \) of a diameter, the slope of the conjugate diameter \( m' \) is given by:
\[ m \cdot m' = -\frac{b^2}{a^2} \]
From this, we can solve for \( m' \):
\[ m' = -\frac{b^2}{a^2 m} \]

### Step 3: Calculate the conjugate diameters
1. **For the first diameter \( x - y = 0 \) (slope \( m_1 = 1 \))**:
\[ m' = -\frac{b^2}{a^2 \cdot 1} = -\frac{b^2}{a^2} \]
The equation of the conjugate diameter is:
\[ y = -\frac{b^2}{a^2} x \quad \text{or} \quad b^2 x + a^2 y = 0 \]

2. **For the second diameter \( x + y = 0 \) (slope \( m_2 = -1 \))**:
\[ m' = -\frac{b^2}{a^2 \cdot (-1)} = \frac{b^2}{a^2} \]
The equation of the conjugate diameter is:
\[ y = \frac{b^2}{a^2} x \quad \text{or} \quad b^2 x - a^2 y = 0 \]

3. **For the third diameter \( y = \frac{a}{b} x \) (slope \( m_3 = \frac{a}{b} \))**:
\[ m' = -\frac{b^2}{a^2 \cdot \frac{a}{b}} = -\frac{b^3}{a^3} \]
The equation of the conjugate diameter is:
\[ y = -\frac{b^3}{a^3} x \quad \text{or} \quad b^3 x + a^3 y = 0 \]

4. **For the fourth diameter \( y = \frac{b}{a} x \) (slope \( m_4 = \frac{b}{a} \))**:
\[ m' = -\frac{b^2}{a^2 \cdot \frac{b}{a}} = -\frac{b^3}{a^3} \]
The equation of the conjugate diameter is:
\[ y = -\frac{a^3}{b^3} x \quad \text{or} \quad a^3 x + b^3 y = 0 \]

### Summary of Conjugate Diameters
1. \( b^2 x + a^2 y = 0 \)
2. \( b^2 x - a^2 y = 0 \)
3. \( b^3 x + a^3 y = 0 \)
4. \( a^3 x + b^3 y = 0 \)