The combined equation of two adjacent sides of a rhombus formed in first quadrant is `7x^2-8xy+y^2=0` then slope of its longer diagonal is

Options

A. `-(1)/(2)`
B. -2
C. 2
D. `(1)/(2)`

Answer

To find the slope of the longer diagonal of the rhombus formed by the given equation of the pair of straight lines, we can follow these steps:

### Step 1: Identify the given equation
The combined equation of the two adjacent sides of the rhombus is given as:
\[ 7x^2 - 8xy + y^2 = 0 \]

### Step 2: Factor the equation
We can factor the quadratic equation to find the two lines. We will use the middle-term splitting method:
\[ 7x^2 - 8xy + y^2 = 0 \]
This can be rewritten as:
\[ 7x^2 - 7xy - xy + y^2 = 0 \]
Grouping the terms:
\[ 7x(x - y) - y(x - y) = 0 \]
Factoring out \( (x - y) \):
\[ (x - y)(7x - y) = 0 \]

### Step 3: Find the equations of the lines
From the factored form, we have two lines:
1. \( x - y = 0 \) or \( y = x \) (Line 1)
2. \( 7x - y = 0 \) or \( y = 7x \) (Line 2)

### Step 4: Determine the slopes of the lines
The slopes of the two lines are:
- For Line 1: \( m_1 = 1 \) (slope of \( y = x \))
- For Line 2: \( m_2 = 7 \) (slope of \( y = 7x \))

### Step 5: Find the angles corresponding to the slopes
Let \( \theta_1 \) be the angle corresponding to \( m_1 \) and \( \theta_2 \) be the angle corresponding to \( m_2 \):
- \( \tan(\theta_1) = 1 \) implies \( \theta_1 = 45^\circ \)
- \( \tan(\theta_2) = 7 \) implies \( \theta_2 = \tan^{-1}(7) \)

### Step 6: Calculate the angle between the two lines
The angle \( \theta \) between the two lines can be found using the formula:
\[ \tan(\theta) = \left| \frac{m_2 - m_1}{1 + m_1 m_2} \right| \]
Substituting the values:
\[ \tan(\theta) = \left| \frac{7 - 1}{1 + (1)(7)} \right| = \left| \frac{6}{8} \right| = \frac{3}{4} \]

### Step 7: Find the slope of the longer diagonal
The slope of the longer diagonal can be found using the angle bisector theorem. The slope of the diagonal \( m_d \) can be calculated using:
\[ \tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)} \]
Substituting \( \tan(\theta) = \frac{3}{4} \):
\[ \tan(2\theta) = \frac{2 \cdot \frac{3}{4}}{1 - \left(\frac{3}{4}\right)^2} = \frac{\frac{3}{2}}{1 - \frac{9}{16}} = \frac{\frac{3}{2}}{\frac{7}{16}} = \frac{3 \cdot 16}{2 \cdot 7} = \frac{48}{14} = \frac{24}{7} \]

### Step 8: Determine the slope of the longer diagonal
The slope of the longer diagonal is:
\[ m_d = 2 \]

Thus, the slope of the longer diagonal of the rhombus is **2**.