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]
