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Determine the angle of rotation needed to eliminate the [tex]xy[/tex] term from the equation [tex]x^2 - xy + y^2 = 0[/tex].

Sagot :

Certainly! Let's find the angle of rotation needed to eliminate the [tex]\(xy\)[/tex] term from the given equation:

[tex]\[ x^2 - xy + y^2 = 0 \][/tex]

The general form of a second-degree equation involving [tex]\(x\)[/tex] and [tex]\(y\)[/tex] is:

[tex]\[ Ax^2 + 2Hxy + By^2 + \dots = 0 \][/tex]

For our equation, we identify:

[tex]\[ A = 1, \quad 2H = -1, \quad B = 1 \][/tex]

First, we need to find the angle [tex]\(\theta\)[/tex] of rotation that will eliminate the [tex]\(xy\)[/tex] term. The relationship between the coefficients and the angle [tex]\(\theta\)[/tex] for rotation is given by:

[tex]\[ \tan(2\theta) = \frac{2H}{A - B} \][/tex]

Given our values:

[tex]\[ A = 1 \][/tex]
[tex]\[ B = 1 \][/tex]
[tex]\[ 2H = -1 \][/tex]

We substitute these into the equation for [tex]\(\tan(2\theta)\)[/tex]:

[tex]\[ \tan(2\theta) = \frac{2H}{A - B} = \frac{-1}{1 - 1} \][/tex]

Notice that [tex]\(A - B = 0\)[/tex]. Therefore, we have:

[tex]\[ \tan(2\theta) = \frac{-1}{0} \][/tex]

Since division by zero is undefined, this implies that [tex]\(\tan(2\theta)\)[/tex] is undefined. The tangent function is undefined at odd multiples of [tex]\( \frac{\pi}{2} \)[/tex] radians (90 degrees).

Consequently, the angle [tex]\(2\theta\)[/tex] must be:

[tex]\[ 2\theta = \frac{\pi}{2} \][/tex]

Solving for [tex]\(\theta\)[/tex]:

[tex]\[ \theta = \frac{\pi}{4} \quad \text{or} \quad \theta = 45^\circ \][/tex]

Additionally:

[tex]\[ 2\theta = -\frac{\pi}{2} \][/tex]

[tex]\[ \theta = -\frac{\pi}{4} \quad \text{or} \quad \theta = -45^\circ \][/tex]

Thus, the angles of rotation that will eliminate the [tex]\(xy\)[/tex] term from the given equation are:

[tex]\[ \theta = 45^\circ \quad \text{or} \quad \theta = -45^\circ \][/tex]

Therefore, the required angles of rotation are:

[tex]\[ \boxed{45^\circ \text{ or } -45^\circ} \][/tex]