Rotate the axes to eliminate the x y -term in the equation. Then write the equation in standard

step 1 In this problem of here rotation of conics we have given the equation that is xY this is xY plus

Rotate the axes to eliminate the x y -term in the equation....

Key Concepts

Graphing Conic Sections with Rotated Axes

After transforming the equation to its standard form via rotation, one can graph the conic section in the new (u, v) coordinate system while also showing the original (x, y) axes. This dual representation helps in understanding how the rotation affects the conic and provides insight into its geometric nature.

Elimination of the Cross-Product Term

Once the axes are rotated appropriately, the cross-product (xy) term in the quadratic equation vanishes. This elimination is crucial for rewriting the equation in a simplified or standard form, which facilitates identifying the type of conic section involved.

Standard Form of Conic Sections

Rewriting the equation in standard form (for example, forms corresponding to ellipses, hyperbolas, or parabolas) allows for an easier analysis of its geometric properties such as center, axes, vertices, and foci. The standard form is achieved by completing the transformation and often by completing the square in the new coordinate variables.

Determination of the Angle of Rotation

This involves finding the specific angle by which the coordinate axes should be rotated in order to remove the xy-term from the conic section’s equation. The angle is typically determined using the formula tan(2?) = B/(A - C), where A, B, and C are coefficients from the general quadratic equation.

Rotation of Axes

This concept refers to changing the coordinate system by turning the axes through a certain angle. The purpose is often to simplify a conic section’s equation by eliminating the cross-product term (xy-term), which can make classification and graphing more straightforward.

Coordinate Transformation Formulas

After determining the rotation angle, new variables are introduced through transformation formulas such as x = u cos ? - v sin ? and y = u sin ? + v cos ?. These formulas express the old coordinates in terms of the new rotated coordinates and are essential for rewriting the equation without the mixed term.

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