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 it is rotated 45 degrees, you can follow these steps:
Step 1: Rewrite the original equation in terms of x' and y' using a rotation matrix. The rotation matrix for a counterclockwise rotation of 45 degrees is:
[cos(45) -sin(45)] [x']
[sin(45) cos(45)] * [y']
Simplifying this expression gives you:
[(√2/2) (-√2/2)] [x']
[(√2/2) (√2/2)] [y']
Step 2: Substitute x' and y' into the original equation. The original equation is xy = 4. Replacing x with (√2/2)x' - (√2/2)y' and y with (√2/2)x' + (√2/2)y', you get:
[(√2/2)x' - (√2/2)y'] * [(√2/2)x' + (√2/2)y'] = 4
Simplifying this expression gives you:
[(√2/2)x']² - [(√2/2)y']² = 4
Step 3: Simplify the equation. In this case, you can multiply (√2/2)² with x'² and y'² to get:
(x')²/2 - (y')²/2 = 4
This gives you:
(x')² - (y')² = 8
Therefore, the correct choice is (C) - (x')² - (y')² = 8.
To find the transformed equation of the hyperbola xy = 4 when rotated 45 degrees, we need to apply a rotation transformation to the original equation.
1. Start by converting the original equation to standard form. Divide both sides of the equation by 4 to get xy/4 = 1.
2. Next, we make use of the following rotation formula for a point (x, y) by an angle of θ:
x' = x*cos(θ) - y*sin(θ)
y' = x*sin(θ) + y*cos(θ)
3. Substitute the values of x and y from the original equation into the rotation formulas, where θ is 45 degrees:
x' = x*cos(45°) - y*sin(45°)
y' = x*sin(45°) + y*cos(45°)
4. Simplify these equations using the values of cos(45°) = sin(45°) = √2 / 2:
x' = √2/2 * x - √2/2 * y
y' = √2/2 * x + √2/2 * y
5. Substitute x' and y' back into the original equation:
(√2/2 * x - √2/2 * y)(√2/2 * x + √2/2 * y) = 4
Simplifying this expression gives us (√2 * x - √2 * y)(√2 * x + √2 * y) = 4.
6. Expanding the equation further yields:
(2x - 2y)(2x + 2y) = 4
4x² - 4y² = 4
Now, compare this transformed equation with the given choices:
A - (x')²(y')² = 4
B - (x')² - (y')² = 4
C - (x')² - (y')² = 8
We can see that the transformed equation matches option B, which is (x')² - (y')² = 4.