How do i find the transformed equation of the hyperbola xy = 4 when rotated 45 degrees? I don't understand at all.
my choices are:
A - (x')²(y')² = 4
B - (x')² - (y')² = 4
C - (x')² - (y')² = 8
To find the transformed equation of the hyperbola xy = 4 when rotated 45 degrees, we need to go through the following steps:
Step 1: Understand the general form of a hyperbola equation.
The general form of a hyperbola equation is given by `{(x-h)²/a²} - {(y-k)²/b²} = 1` for a horizontal hyperbola, and vice versa for a vertical hyperbola.
Step 2: Identify the given equation.
The given equation is `xy = 4`.
Step 3: Determine the standard form of the equation.
Since the equation `xy = 4` is not in standard form, we need to transform it. Divide the equation by 4 to get `xy/4 = 1`.
Step 4: Rotate the axes by 45 degrees.
Rotating the axes by 45 degrees means we need to use a rotation matrix. In this case, the rotation matrix is:
```
cos(θ) -sin(θ)
sin(θ) cos(θ)
```
Where θ is the angle of rotation, which is 45 degrees in this case. So the rotation matrix becomes:
```
cos(45) -sin(45)
sin(45) cos(45)
```
Simplifying this matrix, we get:
```
√2/2 -√2/2
√2/2 √2/2
```
Step 5: Apply the rotation matrix to the equation.
To apply the rotation matrix to the equation `xy/4 = 1`, we multiply both sides of the equation by the rotation matrix:
```
([√2/2 -√2/2] [x]) ([√2/2 -√2/2] [y]) / 4 = 1
([x']) ([y'])
```
This simplifies to:
```
(x' - y')² / 2 = 4
```
Step 6: Rewrite the equation in standard form.
To rewrite the equation `(x' - y')² / 2 = 4` in standard form, we need to multiply both sides by 2:
```
(x' - y')² = 8
```
From the above steps, we can see that the transformed equation of the hyperbola xy = 4 when rotated 45 degrees is `(x' - y')² = 8`.
Therefore, the correct choice is option C - `(x')² - (y')² = 8`.