If the acute angle between the lines `x^(2)-4hxy+3y^(2)=0` is `60^(@)` then `h=`

Options

A. `+-(sqrt(5))/2`
B. `+-(sqrt(10))/2`
C. `+-(sqrt(15))/2`
D. `1`

Explanation

To solve the problem, we need to find the value of \( h \) given that the acute angle between the lines represented by the equation \( x^2 - 4hxy + 3y^2 = 0 \) is \( 60^\circ \).

### Step-by-Step Solution:

1. **Identify the coefficients**:
We can rewrite the equation in the standard form of a pair of straight lines:
\[ Ax^2 + 2Hxy + By^2 = 0 \]
Here, \( A = 1 \), \( 2H = -4h \) (so \( H = -2h \)), and \( B = 3 \).

2. **Use the formula for the angle between two lines**:
The formula for the tangent of the angle \( \theta \) between the two lines is given by:
\[ \tan \theta = \frac{2\sqrt{H^2 - AB}}{A + B} \]
Given that \( \theta = 60^\circ \), we have:
\[ \tan 60^\circ = \sqrt{3} \]

3. **Substitute values into the formula**:
Substitute \( A = 1 \), \( H = -2h \), and \( B = 3 \) into the formula:
\[ \sqrt{3} = \frac{2\sqrt{(-2h)^2 - (1)(3)}}{1 + 3} \]
Simplifying the denominator:
\[ \sqrt{3} = \frac{2\sqrt{4h^2 - 3}}{4} \]

4. **Cross-multiply to eliminate the fraction**:
Multiply both sides by 4:
\[ 4\sqrt{3} = 2\sqrt{4h^2 - 3} \]

5. **Divide both sides by 2**:
\[ 2\sqrt{3} = \sqrt{4h^2 - 3} \]

6. **Square both sides**:
\[ (2\sqrt{3})^2 = (4h^2 - 3) \]
This gives:
\[ 12 = 4h^2 - 3 \]

7. **Rearrange the equation**:
Add 3 to both sides:
\[ 15 = 4h^2 \]

8. **Solve for \( h^2 \)**:
Divide both sides by 4:
\[ h^2 = \frac{15}{4} \]

9. **Take the square root**:
\[ h = \pm \sqrt{\frac{15}{4}} = \pm \frac{\sqrt{15}}{2} \]

### Final Answer:
Thus, the values of \( h \) are:
\[ h = \frac{\sqrt{15}}{2} \quad \text{or} \quad h = -\frac{\sqrt{15}}{2} \]