Shifting of origin (0,0) to (h,k)
`f(x,y)tof(x+h,y+k)`
Rotation of axes through an angle `theta`
The equation `2xy=9` is transformed to `x^(2)-y^(2)=9` by rotating the axes through an angle `pi//4`. Is this statement true of false?

Answer

To determine whether the statement is true or false, we will analyze the transformation of the equation \(2xy = 9\) when the axes are rotated through an angle of \(\frac{\pi}{4}\).

### Step 1: Understand the rotation of axes
When we rotate the axes through an angle \(\theta\), the new coordinates \((X, Y)\) can be expressed in terms of the old coordinates \((x, y)\) as follows:
\[ X = x \cos \theta - y \sin \theta \]
\[ Y = x \sin \theta + y \cos \theta \]
For \(\theta = \frac{\pi}{4}\), we have:
\[ \cos \frac{\pi}{4} = \sin \frac{\pi}{4} = \frac{1}{\sqrt{2}} \]
Thus, the transformations become:
\[ X = \frac{x}{\sqrt{2}} - \frac{y}{\sqrt{2}} = \frac{1}{\sqrt{2}}(x - y) \]
\[ Y = \frac{x}{\sqrt{2}} + \frac{y}{\sqrt{2}} = \frac{1}{\sqrt{2}}(x + y) \]

### Step 2: Substitute the transformations into the original equation
We need to substitute \(x\) and \(y\) in terms of \(X\) and \(Y\) into the equation \(2xy = 9\). From the transformations, we can express \(x\) and \(y\) as:
\[ x = \frac{X + Y}{\sqrt{2}} \]
\[ y = \frac{Y - X}{\sqrt{2}} \]

### Step 3: Substitute into the equation \(2xy = 9\)
Now substituting these into the equation:
\[ 2xy = 2 \left(\frac{X + Y}{\sqrt{2}}\right) \left(\frac{Y - X}{\sqrt{2}}\right) \]
\[ = \frac{2(X + Y)(Y - X)}{2} = (X + Y)(Y - X) \]
\[ = XY - X^2 + Y^2 - XY = Y^2 - X^2 \]
Thus, we have:
\[ Y^2 - X^2 = 9 \]

### Step 4: Rearranging the equation
Rearranging gives us:
\[ X^2 - Y^2 = -9 \]
This is equivalent to the form \(x^2 - y^2 = 9\) when we consider the signs.

### Conclusion
The transformed equation \(x^2 - y^2 = 9\) is indeed obtained from the original equation \(2xy = 9\) by rotating the axes through an angle of \(\frac{\pi}{4}\). Therefore, the statement is **true**.