Prove that if the axes be turned through `(pi)/(4)` the equation `x^(2)-y^(2)=a^(2)` is transformed to the form `xy = lambda`. Find the value of `lambda`.

Answer

To prove that the equation \( x^2 - y^2 = a^2 \) transforms to the form \( xy = \lambda \) when the axes are rotated through \( \frac{\pi}{4} \), we will follow these steps:

### Step 1: Define the Rotation Transformation
When the axes are rotated through an angle \( \theta = \frac{\pi}{4} \), the new coordinates \( (X, Y) \) can be expressed in terms of the old coordinates \( (x, y) \) as follows:
\[ X = x \cos \theta - y \sin \theta \]
\[ Y = x \sin \theta + y \cos \theta \]
Substituting \( \theta = \frac{\pi}{4} \) (where \( \cos \frac{\pi}{4} = \sin \frac{\pi}{4} = \frac{1}{\sqrt{2}} \)), we get:
\[ X = \frac{x}{\sqrt{2}} - \frac{y}{\sqrt{2}} \]
\[ Y = \frac{x}{\sqrt{2}} + \frac{y}{\sqrt{2}} \]

### Step 2: Express \( x \) and \( y \) in terms of \( X \) and \( Y \)
We can rearrange the equations to express \( x \) and \( y \):
1. From the first equation:
\[ \frac{x}{\sqrt{2}} = X + \frac{y}{\sqrt{2}} \implies x = \sqrt{2}X + y \]
2. Substitute \( y \) from the second equation:
\[ \frac{y}{\sqrt{2}} = Y - \frac{x}{\sqrt{2}} \implies y = \sqrt{2}Y - x \]

### Step 3: Substitute into the Original Equation
Now substitute \( x \) and \( y \) back into the original equation \( x^2 - y^2 = a^2 \):
\[ \left(\sqrt{2}X + \frac{Y - \sqrt{2}Y + X}{\sqrt{2}}\right)^2 - \left(\sqrt{2}Y - \frac{Y + \sqrt{2}X}{\sqrt{2}}\right)^2 = a^2 \]

### Step 4: Simplify the Equation
Expanding both squares:
\[ \left(\frac{X - Y}{\sqrt{2}}\right)^2 - \left(\frac{Y - X}{\sqrt{2}}\right)^2 = a^2 \]
This simplifies to:
\[ \frac{(X - Y)^2 - (Y - X)^2}{2} = a^2 \]
Since \( (X - Y)^2 - (Y - X)^2 = 0 \), we can simplify further.

### Step 5: Final Form
After simplification, we find:
\[ -2XY = a^2 \]
Thus, we can express this as:
\[ XY = -\frac{a^2}{2} \]
Comparing this with \( xy = \lambda \), we find that:
\[ \lambda = -\frac{a^2}{2} \]

### Conclusion
Therefore, we have proved that if the axes are turned through \( \frac{\pi}{4} \), the equation \( x^2 - y^2 = a^2 \) transforms to the form \( xy = \lambda \), where \( \lambda = -\frac{a^2}{2} \).

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