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

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

Key Concepts

Standard Form of Conic Sections

Expressing a conic equation in standard form is crucial for identifying its type (ellipse, hyperbola, or parabola) and its geometric properties such as center, axes lengths, foci, and vertices. The standard form provides a clear and concise representation that simplifies both analysis and graphing.

Completing the Square

Completing the square is a method used to rewrite a quadratic expression as a perfect square plus or minus a constant. This process is essential for converting conic equations into their standard forms, as it reveals the translation of the graph and the values of key parameters like the center or vertex.

Rotation of Axes

Rotation of axes is a technique used to eliminate the cross-product (xy) term in a conic section’s equation by redefining the coordinate system. By rotating the axes by an appropriate angle, the equation is transformed so that the conic is aligned with the new axes, making it easier to analyze and graph.

Angle of Rotation Determination

To determine the angle of rotation needed to eliminate the xy term, one uses the relation tan(2?) = B/(A - C), where A, B, and C are the coefficients of x², xy, and y² respectively in the quadratic equation. This formula gives the precise rotation required to simplify the conic equation.

Coordinate Transformation

Once the angle of rotation is known, the new coordinates (often labeled X and Y) are established using the transformation formulas x = X cos? - Y sin? and y = X sin? + Y cos?. This step re-expresses the equation in terms of the new coordinate system, where the mixed term is absent.

Transcript

00:01 In this problem of rotation of conics, we have to eliminate the xy term and then write the equation in a standard form and also we have to sketch the curve.

00:09 So we have given the equation xy minus 8x minus this is minus 8x minus 4y equals to 0.

00:18 And now we have to write it in the standard form eliminating this xy term.

00:23 So first we have to compare it with the standard 2 degree equation that is x square plus bxy plus c y plus c y.

00:31 Square plus d x plus e y plus f equals to 0 so now compare it so here a equals to 0 and b equal to this is the coefficient of x y that is here 1 and c equals to 0 so first we have to find the value of chord 2 theta so chord 2 theta is equals to a minus c divided with b so this would be 0 minus 0 divide with 1 so that means 0 and core 2 theta that means here cot this is pi divide with 2 so that means 2 theta equals to pi divide with 2 that means theta equals to pi divide with 4 and now when we convert it so the here x is replaced with this is x -d s cos theta minus y -dess sine theta and y is converted with here x -dess sine theta plus y -dess cost theta and now sine theta equals to cost theta equals to this is here sine pi divide with 4 is equal to cos pi divided with 4 equals to 1 divide with root 2 so this would be here x -dess minus y -dice so this is x -dess minus y -dest root 2 and this is here x -dess plus y -daz divided with root and now the equation we have given is xy minus 8 x minus 4 y equals to 0 so this would be here x y so that means here x square minus y square divided with 2 and this would be minus 8 x equals to this is here x -d s minus y -dash divide with root 2 and this would be minus 4 y equals to x tis plus y dash with root 2 equals to 0.

02:27 And now we have to make the square.

02:29 So this would be here.

02:35 This can be written as here x -des, so this would be x.

02:39 This is x -dice.

02:41 So this is x -dice.

02:42 X -dice divided with root 2 whole square...

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