Shifting or origin (0,0) to (b,k)
`f(x,y)impliesf(x+h,y+k)`
Rotation of axes through an angle `theta`
`f(x,y)impliesf(x cos theta-y sin theta,x sin theta+ycos theta)`
Now Answer the following questions:
The equation `2xy=9` is transformed to `x^(2)-y^(2)=9` by rotating the axes through an anlge `pi//4`, is this statement true of false?

Answer

To determine if the statement "The equation \(2xy=9\) is transformed to \(x^2-y^2=9\) by rotating the axes through an angle \(\frac{\pi}{4}\)" is true or false, we can follow these steps:

### 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:
\[ x' = x \cos \theta - y \sin \theta \]
\[ y' = x \sin \theta + y \cos \theta \]

### Step 2: Substitute \(\theta = \frac{\pi}{4}\)
For \(\theta = \frac{\pi}{4}\), we have:
\[ \cos \frac{\pi}{4} = \sin \frac{\pi}{4} = \frac{1}{\sqrt{2}} \]
Thus, the transformations become:
\[ x' = \frac{1}{\sqrt{2}}(x - y) \]
\[ y' = \frac{1}{\sqrt{2}}(x + y) \]

### Step 3: Substitute into the Original Equation
We start with the original equation:
\[ 2xy = 9 \]
Now, we will express \(xy\) in terms of \(x'\) and \(y'\):
\[ xy = \left(\frac{1}{\sqrt{2}}(x' + y')\right)\left(\frac{1}{\sqrt{2}}(y' - x')\right) \]
Expanding this:
\[ xy = \frac{1}{2}((x' + y')(y' - x')) = \frac{1}{2}(y'^2 - x'^2) \]

### Step 4: Substitute Back into the Original Equation
Now substituting back into the original equation:
\[ 2 \cdot \frac{1}{2}(y'^2 - x'^2) = 9 \]
This simplifies to:
\[ y'^2 - x'^2 = 9 \]

### Step 5: Rearranging the Equation
Rearranging gives us:
\[ x'^2 - y'^2 = -9 \]
This means that the original equation \(2xy = 9\) transforms into:
\[ x^2 - y^2 = 9 \]
after the rotation.

### Conclusion
Thus, the statement "The equation \(2xy=9\) is transformed to \(x^2-y^2=9\) by rotating the axes through an angle \(\frac{\pi}{4}\)" is **true**.