what is an equation of the conic section x^2+y^2=16 after a rotation of 45 degrees about the origin.
A. (x')^2+(y')^2=16
B. (x')^2- (y')^2=16
C. (x')^2- 2x'y'+(y')^2=16
D. (x')^2+ 2x'y'+(y')^2=16
think. It's a circle. Invariant under rotation about its center.
To find the equation of the conic section after a rotation of 45 degrees about the origin, we can use a rotation matrix.
The rotation matrix for a 2D rotation of θ degrees counterclockwise about the origin is given by:
| cos(θ) -sin(θ) |
| sin(θ) cos(θ) |
In this case, θ = 45 degrees.
So, we can apply this rotation matrix to the conic section equation x^2 + y^2 = 16 to obtain the equation in the rotated coordinate system.
Let's substitute the coordinates (x, y) in the original equation with the coordinates (x', y') in the rotated coordinate system.
The equation becomes:
(x')^2 + (y')^2 = 16
This means the correct equation of the conic section after a rotation of 45 degrees about the origin is A. (x')^2 + (y')^2 = 16.