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

Options

A. `(pi)/4`
B. `(pi)/2`
C. `(pi)/3`
D. `(2pi)/(3) `

Answer

To find the angle between the pair of straight lines represented by the equation

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

we can follow these steps:

### Step 1: Rewrite the Equation
We start with the given equation:

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

Using the identity \( \cos^2 \theta + \sin^2 \theta = 1 \), we can rewrite \( \cos^2 \theta - 1 \) as \( -\sin^2 \theta \):

\[ y^2 \sin^2 \theta - xy \sin^2 \theta - x^2 \sin^2 \theta = 0. \]

### Step 2: Factor Out \(\sin^2 \theta\)
We can factor out \(\sin^2 \theta\) from the equation:

\[ \sin^2 \theta (y^2 - xy - x^2) = 0. \]

Since \(\sin^2 \theta\) cannot be zero (as it would imply \(\theta\) is a multiple of \(\pi\), which is not relevant here), we focus on the quadratic in \(y\):

\[ y^2 - xy - x^2 = 0. \]

### Step 3: Identify Coefficients
The quadratic equation can be compared with the standard form \(Ay^2 + By + C = 0\):

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

### Step 4: Calculate the Angle Between the Lines
The angle \(\theta\) between the two lines represented by the quadratic can be calculated using the formula:

\[ \tan \theta = \frac{2\sqrt{AB}}{A + B}. \]

Here, \(A = 1\) and \(B = -x\), so we calculate:

\[ \tan \theta = \frac{2\sqrt{1 \cdot (-x)}}{1 + (-x)} = \frac{2\sqrt{-x}}{1 - x}. \]

### Step 5: Check for Perpendicularity
To check if the lines are perpendicular, we need to see if:

\[ A + B = 0. \]

Substituting our coefficients:

\[ 1 - x = 0 \implies x = 1. \]

Since \(x\) can take any value, we check the condition of perpendicularity directly from the coefficients of \(x^2\) and \(y^2\):

- Coefficient of \(x^2\) is \(-\sin^2 \theta\).
- Coefficient of \(y^2\) is \(\sin^2 \theta\).

Adding these gives:

\[ -\sin^2 \theta + \sin^2 \theta = 0. \]

### Conclusion
Since the sum of the coefficients of \(x^2\) and \(y^2\) is zero, the lines are perpendicular. Therefore, the angle between the pair of lines is:

\[ \text{Angle} = 90^\circ. \]