Use a translation of axes to remove first-degree terms from...

Use a translation of axes to remove first-degree terms from the equation $x y+4 x-8 y+6=0$; that is, find an $x^{\prime} y^{\prime}$-equation for the graph of the given $x y$-equation such that the first-degree terms in $x^{\prime}$ or $y^{\prime}$ have coefficient 0 . (Hint: Replace $x$ and $y$ with $x^{\prime}+h$ and $y^{\prime}+k$, respectively, and find suitable values for $h$ and $k$.)

Key Concepts

Translation of Axes

Translation of axes is a technique used in coordinate geometry where the coordinate system is shifted by a fixed vector. This shift is described by new coordinates, typically defined as x = x' + h and y = y' + k. The purpose of this technique is to simplify the equation of a curve or conic section by moving the origin to a more convenient location, which often makes subsequent analysis or classification more straightforward.

Elimination of Linear Terms

Eliminating linear terms from an equation involves selecting appropriate translation parameters (h and k) to ensure that the coefficients of the first-degree (linear) terms in the new coordinates become zero. This step is crucial when dealing with quadratic equations in two variables, as it transforms a more complicated equation into a form that is easier to analyze, especially in the study of conic sections.

Coordinate Transformation

A coordinate transformation refers to the process of changing from one coordinate system to another. In the context of this problem, the transformation is a translation which shifts every point in the plane by a constant vector (h, k). Such transformations maintain the relative geometry of figures while recasting the algebraic representation of curves, thus facilitating techniques like completing the square or identifying standard forms of conic sections.

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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