Find the measure of acute angle between the lines given by `x^(2) - 4xy + y^(2) = 0`

Answer

To find the measure of the acute angle between the lines given by the equation \( x^2 - 4xy + y^2 = 0 \), we can follow these steps:

### Step 1: Identify the coefficients
The given equation is \( x^2 - 4xy + y^2 = 0 \). We can identify the coefficients as follows:
- \( A = 1 \)
- \( B = 1 \)
- \( 2H = -4 \) (which gives \( H = -2 \))

### Step 2: Use the formula for the angle between two lines
The formula for the tangent of the angle \( \theta \) between two lines represented by the general quadratic equation \( Ax^2 + 2Hxy + By^2 = 0 \) is given by:

\[ \tan \theta = \frac{2\sqrt{H^2 - AB}}{A + B} \]

### Step 3: Calculate \( H^2 - AB \)
First, we calculate \( H^2 \) and \( AB \):
- \( H^2 = (-2)^2 = 4 \)
- \( AB = 1 \cdot 1 = 1 \)

Now, we find \( H^2 - AB \):
\[ H^2 - AB = 4 - 1 = 3 \]

### Step 4: Substitute into the formula
Now we can substitute the values into the formula for \( \tan \theta \):
\[ \tan \theta = \frac{2\sqrt{3}}{1 + 1} = \frac{2\sqrt{3}}{2} = \sqrt{3} \]

### Step 5: Find \( \theta \)
To find \( \theta \), we take the arctangent:
\[ \theta = \tan^{-1}(\sqrt{3}) \]
The angle whose tangent is \( \sqrt{3} \) is \( 60^\circ \).

### Step 6: Conclusion
Since we are looking for the acute angle, the measure of the acute angle between the lines is:
\[ \theta = 60^\circ \]