The angle through which the axes must be rotated, without translation, in anit-closwise sence so that the exprssion `ax^(2)+hxy-by^(2)+2gx+2fy+c` does not contain the mixed product xy, is given by

Options

A. `tan^(-1)((2h)/(a-b))`
B. `(1)/(2)tan^(-1)((2h)/(b-a))`
C. `(1)/(2)tan^(-1)((2h)/(a-b))`
D. `(1)/(2)tan^(-1)((h)/(a-b))`

Explanation

To find the angle through which the axes must be rotated in an anti-clockwise sense so that the expression \( ax^2 + hxy - by^2 + 2gx + 2fy + c \) does not contain the mixed product \( xy \), we will follow these steps:

### Step 1: Understand the Rotation of Axes
We need to rotate the axes by an angle \( \theta \). The transformations for the coordinates are given by:
\[ x' = x \cos \theta - y \sin \theta \]
\[ y' = x \sin \theta + y \cos \theta \]

### Step 2: Substitute the Transformed Coordinates
Substituting \( x' \) and \( y' \) into the expression \( ax^2 + hxy - by^2 + 2gx + 2fy + c \):
\[ x^2 = (x' \cos \theta + y' \sin \theta)^2 \]
\[ y^2 = (y' \cos \theta - x' \sin \theta)^2 \]
\[ xy = (x' \cos \theta + y' \sin \theta)(y' \cos \theta - x' \sin \theta) \]

### Step 3: Expand the Expression
After substituting, we will expand the expression and collect the coefficients of \( x'y' \) (which corresponds to the mixed product \( xy \)).

### Step 4: Find the Coefficient of \( x'y' \)
The coefficient of \( x'y' \) will be derived from the expansion. We need to ensure that this coefficient equals zero to eliminate the mixed product \( xy \).

### Step 5: Set the Coefficient to Zero
The coefficient of \( xy \) after simplification will be:
\[ -2a \cos \theta \sin \theta + 2h \cos^2 \theta - 2b \sin^2 \theta = 0 \]
This can be simplified to:
\[ -2a \sin 2\theta + 2h \cos 2\theta - 2b \sin 2\theta = 0 \]

### Step 6: Rearranging the Equation
Rearranging gives:
\[ (b - a) \sin 2\theta + h \cos 2\theta = 0 \]

### Step 7: Solve for \( \theta \)
This can be rearranged to:
\[ \tan 2\theta = \frac{2h}{a - b} \]
Thus, we find:
\[ 2\theta = \tan^{-1}\left(\frac{2h}{a - b}\right) \]
Therefore,
\[ \theta = \frac{1}{2} \tan^{-1}\left(\frac{2h}{a - b}\right) \]

### Final Answer
The angle through which the axes must be rotated is:
\[ \theta = \frac{1}{2} \tan^{-1}\left(\frac{2h}{a - b}\right) \]