The straight lines represented by the equation `135 x^2-136 x y+33 y^2=0` are equally inclined to the line (a) `x-2y=7` (b) x+2y=7 (c) `x-2y=4` (d) `3x+2y=4`

Options

A. `x-2y=7`
B. `x+2y=7`
C. `x-2y=4`
D. `3x+2y=4`

Explanation

To determine which line is equally inclined to the pair of straight lines represented by the equation \(135x^2 - 136xy + 33y^2 = 0\), we can follow these steps:

### Step 1: Identify the coefficients
The given equation of the pair of lines is:
\[ 135x^2 - 136xy + 33y^2 = 0 \]
From this equation, we can identify the coefficients:
- \(A = 135\)
- \(B = -136\)
- \(C = 33\)

### Step 2: Calculate the angle bisector equation
The angle bisector of the pair of lines can be found using the formula:
\[ \frac{x^2 - y^2}{A - C} = \frac{xy}{H} \]
where \(H\) is the coefficient of \(xy\), which is \(B\).

Substituting the values:
\[ \frac{x^2 - y^2}{135 - 33} = \frac{xy}{-136} \]
This simplifies to:
\[ \frac{x^2 - y^2}{102} = \frac{xy}{-136} \]

### Step 3: Cross-multiply to simplify
Cross-multiplying gives us:
\[ -136(x^2 - y^2) = 102xy \]
This can be rearranged to:
\[ -136x^2 + 136y^2 + 102xy = 0 \]

### Step 4: Rearranging the equation
Rearranging the equation, we can write:
\[ 2x^2 + 3xy - 2y^2 = 0 \]

### Step 5: Factor the quadratic
Factoring the quadratic equation:
\[ (2x - y)(x + 2y) = 0 \]
This gives us the two lines:
1. \(2x - y = 0\) or \(y = 2x\)
2. \(x + 2y = 0\) or \(y = -\frac{1}{2}x\)

### Step 6: Identify the angle bisector
The angle bisector can be represented as:
\[ x + 2y = 0 \]
This line is parallel to the line \(x + 2y = 7\).

### Step 7: Conclusion
Since the angle bisector is parallel to the line \(x + 2y = 7\), we conclude that this line is equally inclined to the given pair of lines.

Thus, the correct answer is:
**(b) \(x + 2y = 7\)**.

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