Determine the equation of the given conic in xy-coordinates when the coordinate axes are rotated through the indicated angle.
xy = x + y, ϕ = π/4

-X² + Y² = 2√2 Y

Explanation:

xy = x + y

When rotated to a XY coordinate system:

x = X cos θ + Y sin θ

y = -X sin θ + Y cos θ

Given θ = π/4:

x = X (√2/2) + Y (√2/2)

y = -X (√2/2) + Y (√2/2)

x = (√2/2) (X + Y)

y = (√2/2) (-X + Y)

Substitute:

xy = x + y

(√2/2) (X + Y) (√2/2) (-X + Y) = (√2/2) (X + Y) + (√2/2) (-X + Y)

(1/2) (X + Y) (-X + Y) = (√2/2) (X + Y − X + Y)

(1/2) (-X² + Y²) = √2 Y

-X² + Y² = 2√2 Y

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