Write the appropriate rotation formulas so that in a rotated system the equation has no x^' y^'

step 1 We have given an equation that is x square plus 4 xY minus 2y square minus 1 equal to 0. We have

Write the appropriate rotation formulas so that in a rotated...

Key Concepts

Determination of the Rotation Angle

The determination of the rotation angle is achieved through the formula tan2? = B/(A - C). Once tan2? is calculated, the angle ? is chosen so that the transformation of coordinates through the rotation transformation equations results in a new quadratic equation devoid of the xy-term. This step is crucial in simplifying the analysis of conic sections.

Elimination of the Cross Product Term

When an xy-term is present in a quadratic equation, it signifies that the conic is not aligned with the coordinate axes. To eliminate this term, one selects a rotation angle such that the coefficient of the x'y' term in the rotated equation becomes zero. This elimination is achieved by using the condition tan2? = B/(A - C), where A, B, and C are the coefficients of x², xy, and y² respectively.

Rotation of Axes

Rotation of axes is a technique used to simplify the equation of a conic section by eliminating the xy-term (or cross product term). By choosing a new set of axes that are rotated relative to the original ones, the quadratic form can be transformed to a simpler canonical form, which often makes it easier to classify and analyze the conic section.

Rotation Transformation Equations

The rotation transformation equations define the relationship between the original coordinates (x, y) and the new coordinates (x', y') after a rotation by an angle ?. These equations are given by x = x' cos? - y' sin? and y = x' sin? + y' cos?. They are central to the process of rotating the coordinate system and re-expressing the given equation without the xy-term.

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