The equation `4xy-3x^(2)=a^(2)` become when the axes are turned through an angle `tan^(-1)2` is

Options

A. `x^(2)+4y^(2)=a^(2)`
B. `x^(2)-4y^(2)=a^(2)`
C. `4x^(2)+y^(2)=a^(2)`
D. `4x^(2)-y^(2)=a^(2)`

Answer

To solve the problem, we need to find the new equation of the given equation \(4xy - 3x^2 = a^2\) when the axes are rotated through an angle \(\theta = \tan^{-1}(2)\).

### Step-by-Step Solution:

1. **Determine the angle of rotation**:
\[ \theta = \tan^{-1}(2) \]
From this, we can find \(\sin \theta\) and \(\cos \theta\).
- Using the right triangle definition, we have opposite side = 2 and adjacent side = 1.
- The hypotenuse \(h\) can be calculated as:
\[ h = \sqrt{2^2 + 1^2} = \sqrt{4 + 1} = \sqrt{5} \]
- Therefore,
\[ \sin \theta = \frac{2}{\sqrt{5}}, \quad \cos \theta = \frac{1}{\sqrt{5}} \]

2. **Substitute the transformations for \(x\) and \(y\)**:
The transformations for the coordinates when the axes are rotated are given by:
\[ x' = x \cos \theta + y \sin \theta \]
\[ y' = -x \sin \theta + y \cos \theta \]
Substituting the values of \(\sin \theta\) and \(\cos \theta\):
\[ x' = x \cdot \frac{1}{\sqrt{5}} + y \cdot \frac{2}{\sqrt{5}} = \frac{x + 2y}{\sqrt{5}} \]
\[ y' = -x \cdot \frac{2}{\sqrt{5}} + y \cdot \frac{1}{\sqrt{5}} = \frac{-2x + y}{\sqrt{5}} \]

3. **Substitute into the original equation**:
We need to substitute \(x\) and \(y\) in terms of \(x'\) and \(y'\) into the original equation \(4xy - 3x^2 = a^2\).
- Rearranging the above equations gives:
\[ x = \frac{x' + 2y'}{\sqrt{5}}, \quad y = \frac{-2x' + y'}{\sqrt{5}} \]

4. **Substituting \(x\) and \(y\) into the equation**:
Substitute these into the original equation:
\[ 4\left(\frac{x' + 2y'}{\sqrt{5}}\right)\left(\frac{-2x' + y'}{\sqrt{5}}\right) - 3\left(\frac{x' + 2y'}{\sqrt{5}}\right)^2 = a^2 \]
Simplifying this expression:
\[ = \frac{4}{5}[(x' + 2y')(-2x' + y')] - \frac{3}{5}(x' + 2y')^2 \]

5. **Expanding and simplifying**:
- Expand the first term:
\[ (x' + 2y')(-2x' + y') = -2x'^2 + x'y' - 4xy' + 2y'^2 \]
- Expand the second term:
\[ (x' + 2y')^2 = x'^2 + 4x'y' + 4y'^2 \]
- Combine these results to get the new equation.

6. **Final equation**:
After simplification, we will arrive at the new equation in terms of \(x'\) and \(y'\).