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Home  Limits, Analytic Geometry, and Approximations Rotation of Axes Theorem 1 |
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Theorem 1
THEOREM 1 Given a second degree equation Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 with B nonzero. Rotate the coordinate axes counterclockwise through an angle α for which
Then the equation A1X2 + B1XY + C1Y2 + D1X + E1Y + F1 = 0 with respect to the new coordinate axes X and Y has XY-term B1 = 0. This theorem can be proved as follows. When the rotation equations are substituted and terms collected, the XY coefficient B1 comes out to be B1 = B(cos2 α - sin2 α) - 2(A - C) sin α cos α. From trigonometry, cos2 α — sin2 α = cos (2α), 2 sin α cos α = sin (2α). Thus B1 = B cos (2α) - (A - C) sin (2α). So B1 = 0 if and only if
or
As shown in Figure 5.7.4, α is the angle between the original coordinate axes and the axes of the parabola, ellipse, or hyperbola.
Figure 5.7.4 We are now ready to use rotation of axes to sketch a second degree curve. We illustrate the method for the curve introduced in Example 1.
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Last Update: 2006-11-06