Shifting or origin (0,0) to (b,k)
`f(x,y)impliesf(x+h,y+k)`
Rotation of axes through an angle `theta`
`f(x,y)impliesf(x cos theta-y sin theta,x sin theta+ycos theta)`
Now Answer the following questions:
By rotating the axes through an angle `theta` in anti clockwise direction the equation `f(x,y)=x^(2)-2xy+3y^(2)+4x-4y+1=0`
transforms to the form whcih does not contain the term of y then `theta=135^(@)`

Answer

To solve the problem of transforming the equation \( f(x,y) = x^2 - 2xy + 3y^2 + 4x - 4y + 1 = 0 \) by rotating the axes through an angle \( \theta \) such that the resulting equation does not contain the term \( y \), we will follow these steps:

### Step 1: Substitute the Rotation Transformation
We start by substituting the rotation transformation into the equation. The transformation for rotating the axes through an angle \( \theta \) is given by:
\[ x' = x \cos \theta - y \sin \theta \]
\[ y' = x \sin \theta + y \cos \theta \]
We will replace \( x \) and \( y \) in the original equation with \( x' \) and \( y' \).

### Step 2: Expand the Equation
Substituting \( x' \) and \( y' \) into the equation:
\[ f(x', y') = (x \cos \theta - y \sin \theta)^2 - 2(x \cos \theta - y \sin \theta)(x \sin \theta + y \cos \theta) + 3(x \sin \theta + y \cos \theta)^2 + 4(x \cos \theta - y \sin \theta) - 4(x \sin \theta + y \cos \theta) + 1 = 0 \]
Now, we will expand this equation.

### Step 3: Collect Terms
After expanding, we will collect the coefficients of \( y \) in the equation. The goal is to set the coefficient of \( y \) to zero to eliminate the \( y \) term.

### Step 4: Identify the Coefficient of \( y \)
From the expanded equation, we identify the coefficient of \( y \):
- The term from \( -2(x \cos \theta - y \sin \theta)(x \sin \theta + y \cos \theta) \) contributes \( -2y \cos \theta \) and \( +2y \sin \theta \).
- The term from \( -4(x \sin \theta + y \cos \theta) \) contributes \( -4y \cos \theta \).

Thus, the total coefficient of \( y \) is:
\[ -2 \cos \theta + 2 \sin \theta - 4 \cos \theta = -6 \cos \theta + 2 \sin \theta \]

### Step 5: Set the Coefficient to Zero
To eliminate the \( y \) term, we set the coefficient to zero:
\[ -6 \cos \theta + 2 \sin \theta = 0 \]
This simplifies to:
\[ 3 \cos \theta = \sin \theta \]
Dividing both sides by \( \cos \theta \) (assuming \( \cos \theta \neq 0 \)):
\[ 3 = \tan \theta \]

### Step 6: Find \( \theta \)
To find \( \theta \), we take the arctangent:
\[ \theta = \tan^{-1}(3) \]
However, we need to find \( \theta \) such that the transformation results in the equation not containing \( y \). Given the problem states \( \theta = 135^\circ \), we can verify:
\[ \tan(135^\circ) = -1 \]
This means we need to check if \( \tan(135^\circ) \) satisfies our earlier condition.

### Conclusion
Thus, the angle \( \theta \) that allows the transformation to eliminate the \( y \) term is indeed \( 135^\circ \).