The angle between the pair of lines represented by `(sin^(2)alpha)(x^(2)+y^(2))=(xcosalpha-ysinalpha)^(2)` is

Options

A. `alpha`
B. `2alpha`
C. `-alpha`
D. `-2alpha`

Answer

To find the angle between the pair of lines represented by the equation \((\sin^2 \alpha)(x^2 + y^2) = (x \cos \alpha - y \sin \alpha)^2\), we will follow these steps:

### Step 1: Rewrite the equation
We start with the given equation:
\[ \sin^2 \alpha (x^2 + y^2) = (x \cos \alpha - y \sin \alpha)^2 \]

### Step 2: Expand the right-hand side
Expanding the right-hand side:
\[ (x \cos \alpha - y \sin \alpha)^2 = x^2 \cos^2 \alpha - 2xy \cos \alpha \sin \alpha + y^2 \sin^2 \alpha \]

### Step 3: Substitute the expansion back into the equation
Now we substitute this back into the equation:
\[ \sin^2 \alpha (x^2 + y^2) = x^2 \cos^2 \alpha - 2xy \cos \alpha \sin \alpha + y^2 \sin^2 \alpha \]

### Step 4: Rearrange the equation
Rearranging gives us:
\[ \sin^2 \alpha x^2 + \sin^2 \alpha y^2 - x^2 \cos^2 \alpha + 2xy \cos \alpha \sin \alpha - y^2 \sin^2 \alpha = 0 \]
This simplifies to:
\[ (\sin^2 \alpha - \cos^2 \alpha)x^2 + 2xy \cos \alpha \sin \alpha + (0)y^2 = 0 \]

### Step 5: Identify coefficients
From the equation \(Ax^2 + Bxy + Cy^2 = 0\), we identify:
- \(A = \sin^2 \alpha - \cos^2 \alpha\)
- \(B = 2 \cos \alpha \sin \alpha\)
- \(C = 0\)

### Step 6: Use the formula for the angle between the lines
The formula for the angle \(\theta\) between the lines is given by:
\[ \tan \theta = \frac{2\sqrt{h^2 - AB}}{A + B} \]
where \(h = \frac{B}{2}\).

### Step 7: Calculate \(h^2 - AB\)
Here, \(h = \cos \alpha \sin \alpha\), so:
\[ h^2 = (\cos \alpha \sin \alpha)^2 = \cos^2 \alpha \sin^2 \alpha \]
Now, calculate \(AB\):
\[ AB = (\sin^2 \alpha - \cos^2 \alpha)(2 \cos \alpha \sin \alpha) \]

### Step 8: Substitute into the formula
Substituting into the tangent formula:
\[ \tan \theta = \frac{2 \cdot \cos \alpha \sin \alpha}{\sin^2 \alpha - \cos^2 \alpha} \]

### Step 9: Simplify the expression
Using the double angle identities:
\[ \tan \theta = \tan(2\alpha) \]

### Step 10: Final result
Thus, the angle between the pair of lines is:
\[ \theta = 2\alpha \]