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 of transforming the equation \(4xy - 3x^2 = a^2\) when the axes are rotated through an angle of \(\tan^{-1}(2)\), we will follow these steps:

### Step 1: Determine the angle of rotation
The angle of rotation \(\theta\) is given by:
\[ \theta = \tan^{-1}(2) \]
From this, we can find:
\[ \tan \theta = 2 \]
Using the identity \(\tan \theta = \frac{\sin \theta}{\cos \theta}\), we can derive:
\[ \sin \theta = \frac{2}{\sqrt{5}}, \quad \cos \theta = \frac{1}{\sqrt{5}} \]

### Step 2: Write the transformation equations
The transformation of coordinates from \((x, y)\) to \((X, Y)\) is given by:
\[ x = X \cos \theta - Y \sin \theta = X \cdot \frac{1}{\sqrt{5}} - Y \cdot \frac{2}{\sqrt{5}} = \frac{X - 2Y}{\sqrt{5}} \]
\[ y = X \sin \theta + Y \cos \theta = X \cdot \frac{2}{\sqrt{5}} + Y \cdot \frac{1}{\sqrt{5}} = \frac{2X + Y}{\sqrt{5}} \]

### Step 3: Substitute the transformations into the original equation
We substitute \(x\) and \(y\) into the original equation \(4xy - 3x^2 = a^2\):
\[ 4\left(\frac{X - 2Y}{\sqrt{5}}\right)\left(\frac{2X + Y}{\sqrt{5}}\right) - 3\left(\frac{X - 2Y}{\sqrt{5}}\right)^2 = a^2 \]

### Step 4: Simplify the equation
Calculating \(4xy\):
\[ 4xy = 4 \cdot \frac{(X - 2Y)(2X + Y)}{5} = \frac{4(2X^2 + XY - 4Y^2)}{5} \]
Calculating \(3x^2\):
\[ 3x^2 = 3 \cdot \left(\frac{(X - 2Y)^2}{5}\right) = \frac{3(X^2 - 4XY + 4Y^2)}{5} \]
Now substituting these into the equation:
\[ \frac{4(2X^2 + XY - 4Y^2)}{5} - \frac{3(X^2 - 4XY + 4Y^2)}{5} = a^2 \]
Combining the fractions:
\[ \frac{4(2X^2 + XY - 4Y^2) - 3(X^2 - 4XY + 4Y^2)}{5} = a^2 \]
Simplifying the numerator:
\[ 8X^2 + 4XY - 16Y^2 - 3X^2 + 12XY - 12Y^2 = 0 \]
This simplifies to:
\[ (8X^2 - 3X^2) + (4XY + 12XY) + (-16Y^2 - 12Y^2) = 0 \]
\[ 5X^2 + 16XY - 28Y^2 = 5a^2 \]
Thus, we can write:
\[ 5X^2 + 16XY - 28Y^2 = 5a^2 \]

### Final Result
The transformed equation when the axes are turned through an angle \(\tan^{-1}(2)\) is:
\[ 5X^2 + 16XY - 28Y^2 = 5a^2 \]