If by rotating the axes through an angle `theta` the general equation of second degree
`ax^(2)+2hxy+by^(2)+2gx+2fy+c=0` is free form the term of xy, then `tan 2 theta` is

Options

A. `(a-b)/(2h)`
B. `(2h)/(a+b)`
C. `(a+b)/(2h)`
D. `(2h)/(a-b)`

Answer

To solve the problem, we need to find the value of \( \tan 2\theta \) when the general equation of the second degree is free from the term \( xy \) after rotating the axes through an angle \( \theta \).

### Step-by-Step Solution:

1. **Understanding the General Equation**:
The general equation of the second degree is given by:
\[ ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 \]

2. **Rotation of Axes**:
When we rotate the axes through an angle \( \theta \), the new coordinates \( x' \) and \( y' \) can be expressed in terms of \( x \) and \( y \) as follows:
\[ x' = x \cos \theta - y \sin \theta \]
\[ y' = x \sin \theta + y \cos \theta \]

3. **Substituting New Coordinates**:
Substitute \( x' \) and \( y' \) into the general equation. This involves expanding the terms:
\[ ax'^2 + 2hx'y' + by'^2 + 2gx' + 2fy' + c = 0 \]
After substitution, we will collect the coefficients of \( x'y' \).

4. **Finding the Coefficient of \( x'y' \)**:
The coefficient of \( x'y' \) after substitution will be:
\[ -A \sin 2\theta + B \sin 2\theta + 2h \cos^2 \theta - \sin^2 \theta \]
We need this coefficient to be zero for the equation to be free from the \( x'y' \) term.

5. **Setting the Coefficient to Zero**:
Set the coefficient of \( x'y' \) to zero:
\[ -A + B + 2h \cos^2 \theta - \sin^2 \theta = 0 \]
Rearranging gives:
\[ \sin 2\theta (B - A) + 2h \cos^2 \theta - \sin^2 \theta = 0 \]

6. **Factoring Out \( \sin 2\theta \)**:
Factor out \( \sin 2\theta \):
\[ \sin 2\theta (B - A) = 2h \cos^2 \theta - \sin^2 \theta \]

7. **Using Trigonometric Identities**:
Recall that \( \cos^2 \theta = \frac{1 + \cos 2\theta}{2} \) and \( \sin^2 \theta = \frac{1 - \cos 2\theta}{2} \). Substitute these into the equation.

8. **Finding \( \tan 2\theta \)**:
From the equation:
\[ \tan 2\theta = \frac{2h}{A - B} \]

### Final Result:
Thus, the value of \( \tan 2\theta \) is:
\[ \tan 2\theta = \frac{2h}{A - B} \]