Determine the equation of the given conic in XYcoordinates w

Determine the equation of the given conic in XY-coordinates when the coordinate axes are rotated through the indicated angle.

x² + 2y² = 12,   = sin^{1 (3/5)}

Solution

x^2 + 2y^2 = 12,   = sin1 (3/5) = 36.9 degree

x = Xcos + Ysin = 0.8*X + 0.6*Y
y = -Xsin + Ycos = -0.6*X + 0.8*Y
Substitute = 36.9 degrees and simplify x and y.
Next, substitute x and y to x^2 + 2y^2 = 12

(0.8*X + 0.6*Y)^2 + 2*(-0.6*X + 0.8*Y)^2 = 12

Then simplify, the result will be required.