The transformed equation of `x^(2)+6xy+8y^(2)=10` when the axes are rotated through an angled `pi//4` is

Options

A. `15x^(2)-14xy+3y^(2)=20`
B. `15x^(2)+14xy-3y^(2)=20`
C. `15x^(2)+14xy+3y^(2)=20`
D. `15x^(2)-14xy-3y^(2)=20`

Answer

To find the transformed equation of \( x^2 + 6xy + 8y^2 = 10 \) when the axes are rotated through an angle of \( \frac{\pi}{4} \), we will follow these steps:

### Step 1: Substitute for x and y
When the axes are rotated through an angle \( \theta \), the new coordinates \( x' \) and \( y' \) can be expressed in terms of the old coordinates \( x \) and \( y \) as follows:
\[ x = x' \cos \theta - y' \sin \theta \]
\[ y = x' \sin \theta + y' \cos \theta \]
For \( \theta = \frac{\pi}{4} \), we have \( \cos \frac{\pi}{4} = \sin \frac{\pi}{4} = \frac{1}{\sqrt{2}} \). Thus, we can write:
\[ x = \frac{1}{\sqrt{2}}(x' - y') \]
\[ y = \frac{1}{\sqrt{2}}(x' + y') \]

### Step 2: Substitute into the original equation
We substitute these expressions for \( x \) and \( y \) into the original equation \( x^2 + 6xy + 8y^2 = 10 \).

1. Calculate \( x^2 \):
\[ x^2 = \left(\frac{1}{\sqrt{2}}(x' - y')\right)^2 = \frac{1}{2}(x'^2 - 2x'y' + y'^2) \]

2. Calculate \( y^2 \):
\[ y^2 = \left(\frac{1}{\sqrt{2}}(x' + y')\right)^2 = \frac{1}{2}(x'^2 + 2x'y' + y'^2) \]

3. Calculate \( xy \):
\[ xy = \left(\frac{1}{\sqrt{2}}(x' - y')\right)\left(\frac{1}{\sqrt{2}}(x' + y')\right) = \frac{1}{2}(x'^2 - y'^2) \]

### Step 3: Substitute into the equation
Now we substitute these results into the original equation:
\[ \frac{1}{2}(x'^2 - 2x'y' + y'^2) + 6 \cdot \frac{1}{2}(x'^2 - y'^2) + 8 \cdot \frac{1}{2}(x'^2 + 2x'y' + y'^2) = 10 \]

### Step 4: Simplify the equation
Combining the terms:
\[ \frac{1}{2}(x'^2 - 2x'y' + y'^2) + 3(x'^2 - y'^2) + 4(x'^2 + 2x'y' + y'^2) = 10 \]
\[ = \frac{1}{2}x'^2 - x'y' + \frac{1}{2}y'^2 + 3x'^2 - 3y'^2 + 4x'^2 + 8x'y' + 4y'^2 = 10 \]
Combining like terms:
\[ ( \frac{1}{2} + 3 + 4 )x'^2 + ( \frac{1}{2} - 3 + 4 )y'^2 + ( -1 + 8 )x'y' = 10 \]
\[ 8.5x'^2 + 1.5y'^2 + 7x'y' = 10 \]

### Step 5: Clear the fractions
To eliminate the fractions, multiply through by 2:
\[ 17x'^2 + 3y'^2 + 14x'y' = 20 \]

### Final Answer
The transformed equation after rotating the axes through an angle of \( \frac{\pi}{4} \) is:
\[ 17x'^2 + 3y'^2 + 14x'y' = 20 \]