Findthe angle between the pair of straight lines `x^2 -4y^2+3x - 4 = 0 `

Answer

To find the angle between the pair of straight lines given by the equation \(x^2 - 4y^2 + 3x - 4 = 0\), we can follow these steps:

### Step 1: Rewrite the equation in standard form
The given equation is:
\[ x^2 - 4y^2 + 3x - 4 = 0 \]
We can rearrange it to group the terms:
\[ x^2 + 3x - 4y^2 - 4 = 0 \]

### Step 2: Identify coefficients
In the general form of the equation of a pair of straight lines \(Ax^2 + 2Hxy + By^2 + 2Gx + 2Fy + C = 0\), we can identify:
- \(A = 1\)
- \(B = -4\)
- \(H = 0\) (since there is no \(xy\) term)
- \(G = \frac{3}{2}\) (since \(2G = 3\), hence \(G = \frac{3}{2}\))
- \(F = 0\)
- \(C = -4\)

### Step 3: Use the formula for the angle between the lines
The angle \(\theta\) between the two lines can be found using the formula:
\[ \tan \theta = \frac{2\sqrt{H^2 - AB}}{A + B} \]
Substituting the values we identified:
- \(A = 1\)
- \(B = -4\)
- \(H = 0\)

### Step 4: Calculate \(H^2 - AB\)
First, calculate \(H^2 - AB\):
\[ H^2 - AB = 0^2 - (1)(-4) = 0 + 4 = 4 \]

### Step 5: Substitute into the angle formula
Now substitute into the formula:
\[ \tan \theta = \frac{2\sqrt{4}}{1 - 4} = \frac{2 \times 2}{1 - 4} = \frac{4}{-3} \]
Since we are interested in the angle, we take the positive value:
\[ \tan \theta = \frac{4}{3} \]

### Step 6: Find the angle \(\theta\)
To find the angle \(\theta\), we can use the arctangent function:
\[ \theta = \tan^{-1}\left(\frac{4}{3}\right) \]

### Conclusion
Thus, the angle between the pair of straight lines is:
\[ \theta = \tan^{-1}\left(\frac{4}{3}\right) \]