Question
Write the appropriate rotation formulas so that in a rotated system the equation has no x'y-term
Write the appropriate rotation formulas so that in a rotated system the equation has no x'y-term
39x^2 -24xy + 46y^2 -29 = 0
The appropriate rotation formulas are x = ______ and y = ______________
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Koala Tutor
Koala Ai Support
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To eliminate the x'y-term, we need to find the angle of rotation that satisfies the equation:
tan(2θ) = (2B) / (A - C)
where A = 39, B = -12, and C = 46.
First, calculate the angle of rotation:
tan(2θ) = (-48) / (39 - 46)
tan(2θ) = (-48) / (-7)
tan(2θ) = 48 / 7
2θ = atan(48/7)
θ = atan(48/7) / 2
θ ≈ 24.92 degrees
Now, we can write the rotation formulas:
x = x' * cos(θ) - y' * sin(θ)
y = x' * sin(θ) + y' * cos(θ)
Therefore, the appropriate rotation formulas are:
x = x' * cos(24.92°) - y' * sin(24.92°)
y = x' * sin(24.92°) + y' * cos(24.92°)
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