The angle between the pair of straight lines `y^2sin^2theta-xysintheta+x^2(cos^2theta-1)=0` is

Options

A. `(pi)/(3)`
B. `(pi)/(4)`
C. `(2pi)/(3)`
D. none of these

Explanation

To find the angle between the pair of straight lines given by the equation:

\[ y^2 \sin^2 \theta - xy \sin \theta + x^2 (\cos^2 \theta - 1) = 0 \]

we can follow these steps:

### Step 1: Rewrite the equation
The given equation can be rearranged as:

\[ y^2 \sin^2 \theta - xy \sin \theta + x^2 (\cos^2 \theta - 1) = 0 \]

This is a quadratic equation in terms of \(y\).

### Step 2: Identify coefficients
In the standard form of a quadratic equation \(Ay^2 + By + C = 0\), we identify the coefficients:

- \(A = \sin^2 \theta\)
- \(B = -x \sin \theta\)
- \(C = x^2 (\cos^2 \theta - 1)\)

### Step 3: Use the formula for the angle between two lines
The angle \( \phi \) between the two lines represented by the quadratic equation can be found using the formula:

\[ \tan \phi = \frac{2\sqrt{h^2 - ab}}{a + b} \]

where \(a = A\), \(b = C\), and \(h = B\).

### Step 4: Calculate \(h^2 - ab\)
First, we calculate \(h^2 - ab\):

- \(h = -x \sin \theta\)
- \(a = \sin^2 \theta\)
- \(b = x^2 (\cos^2 \theta - 1)\)

Now, we compute:

\[ h^2 = (-x \sin \theta)^2 = x^2 \sin^2 \theta \]

\[ ab = \sin^2 \theta \cdot x^2 (\cos^2 \theta - 1) = x^2 \sin^2 \theta (\cos^2 \theta - 1) \]

Now substituting these into the formula:

\[ h^2 - ab = x^2 \sin^2 \theta - x^2 \sin^2 \theta (\cos^2 \theta - 1) \]

This simplifies to:

\[ h^2 - ab = x^2 \sin^2 \theta (1 - (\cos^2 \theta - 1)) = x^2 \sin^2 \theta (1 - \cos^2 \theta + 1) = x^2 \sin^2 \theta (2 - \cos^2 \theta) \]

### Step 5: Calculate \(a + b\)
Now we calculate \(a + b\):

\[ a + b = \sin^2 \theta + x^2 (\cos^2 \theta - 1) \]

### Step 6: Determine the angle
To find the angle between the lines, we need to check if the lines are perpendicular. For the lines to be perpendicular, we need:

\[ h^2 - ab = 0 \]

If this condition holds, the angle between the lines is \(90^\circ\).

### Conclusion
After evaluating the conditions, we find that the angle between the pair of straight lines is \(90^\circ\).

### Final Answer
The angle between the pair of straight lines is \(90^\circ\).

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