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 follow a few steps:
Step 1: Write the original equation in terms of x' and y' (the rotated coordinate system).
Step 2: Apply the rotation transformation to the original equation.
Step 3: Simplify the equation and determine the constants involved.
Now, let's break down each step:
Step 1: Writing the original equation in terms of x' and y'
We know that the original equation is xy = 4. To express this equation in terms of the rotated coordinates, we need to use a rotation matrix. The rotation matrix for a 45-degree rotation is:
| √2/2 -√2/2 |
| √2/2 √2/2 |
So, we can write the original equation as:
| x' | | √2/2 -√2/2 | | x |
| | = | | x | |
| y' | | √2/2 √2/2 | | y |
Multiplying these matrices, we get:
x' * y' = (√2/2 * x - √2/2 * y) * (√2/2 * x + √2/2 * y)
x' * y' = (2/2) * (x² - y²)
x' * y' = x² - y²
Step 2: Applying the rotation transformation
By applying the rotation transformation, we obtain the equation:
x' * y' = x² - y²
Step 3: Simplifying and determining the constants
Comparing the transformed equation with the given choices:
A - (x')²(y')² = 4
B - (x')² - (y')² = 4
C - (x')² - (y')² = 8
None of the choices match the simplified transformed equation, x' * y' = x² - y².
Therefore, none of the options (A, B, C) are correct.
To summarize, the transformed equation of the hyperbola xy = 4 when rotated 45 degrees is x' * y' = x² - y².