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 combined equation of two adjacent sides, we will follow these steps:

### Step 1: Write 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 quadratic equation
We will factor the quadratic equation to find the equations of the lines representing the sides of the rhombus. We can rewrite the equation as:
\[ 7x^2 - 8xy + y^2 = 0 \]

Using the middle term splitting method:
\[ 7x^2 - 7xy - xy + y^2 = 0 \]
\[ 7x(x - y) - y(x - y) = 0 \]
Factoring out \( (x - y) \):
\[ (7x - y)(x - y) = 0 \]

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

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

### Step 5: Find the angles corresponding to the slopes
Let \( \theta_1 \) be the angle corresponding to slope \( m_1 = 7 \) and \( \theta_2 \) be the angle corresponding to slope \( m_2 = 1 \).

Using the tangent function:
- \( \tan(\theta_1) = 7 \)
- \( \tan(\theta_2) = 1 \)

### Step 6: Use the angle sum formula for tangents
The angle between the diagonals can be calculated using the formula:
\[ \tan(\theta_1 + \theta_2) = \frac{\tan(\theta_1) + \tan(\theta_2)}{1 - \tan(\theta_1) \tan(\theta_2)} \]
Substituting the values:
\[ \tan(\theta_1 + \theta_2) = \frac{7 + 1}{1 - 7 \cdot 1} = \frac{8}{1 - 7} = \frac{8}{-6} = -\frac{4}{3} \]

### Step 7: Find the slope of the longer diagonal
The slope of the longer diagonal can be found using the relationship between the slopes of the sides and the diagonals. The slope of the diagonal can be calculated using the formula:
\[ \tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)} \]
where \( \tan(\theta) \) is the average of the slopes of the two sides.

Calculating the average slope:
\[ \tan(\theta) = \frac{m_1 + m_2}{2} = \frac{7 + 1}{2} = 4 \]
Now substituting into the formula:
\[ \tan(2\theta) = \frac{2 \cdot 4}{1 - 4^2} = \frac{8}{1 - 16} = \frac{8}{-15} \]

### Step 8: Determine the slope of the longer diagonal
Since we are interested in the longer diagonal, we need to find the slope that corresponds to the acute angle formed by the sides. The slope of the longer diagonal is:
\[ \tan(\theta) = 2 \]

### Final Answer
Thus, the slope of the longer diagonal of the rhombus is:
\[ \text{Slope of longer diagonal} = 2 \]