The angle between the pair of lines whose equation is `4x^(2)+10xy+my^(2)+5x+10y=0`, is

Options

A. `tan^(-1)(3//8)`
B. `tan^(-1)(3//4)`
C. `tan^(-1){2sqrt(25-4m//)(m+4)},m in R`
D. None of these

Explanation

To find the angle between the pair of lines represented by the equation \(4x^2 + 10xy + my^2 + 5x + 10y = 0\), we will follow these steps:

### Step 1: Identify coefficients
The general form of the equation of a pair of straight lines is given by:
\[ ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 \]
From the given equation, we can identify the coefficients:
- \(a = 4\)
- \(b = m\)
- \(c = 0\)
- \(g = 5\)
- \(f = 5\)
- \(h = 5\)

### Step 2: Apply the condition for a pair of straight lines
For the equation to represent a pair of straight lines, the following condition must hold:
\[ abc + 2fgh - a f^2 - b g^2 - c h^2 = 0 \]
Substituting the identified coefficients into this condition:
\[ 4 \cdot m \cdot 0 + 2 \cdot 5 \cdot 5 \cdot 5 - 4 \cdot 5^2 - m \cdot (5/2)^2 - 0 \cdot 5^2 = 0 \]
This simplifies to:
\[ 0 + 250 - 100 - \frac{25m}{4} = 0 \]
Thus, we have:
\[ 250 - 100 - \frac{25m}{4} = 0 \]

### Step 3: Solve for \(m\)
Rearranging the equation gives:
\[ 150 = \frac{25m}{4} \]
Multiplying both sides by 4:
\[ 600 = 25m \]
Dividing by 25:
\[ m = 24 \]

### Step 4: Calculate the angle between the lines
Now that we have \(m\), we can find the angle between the lines using the formula:
\[ \tan \theta = \frac{2\sqrt{h^2 - ab}}{a + b} \]
Substituting \(a = 4\), \(b = 24\), and \(h = 5\):
\[ \tan \theta = \frac{2\sqrt{5^2 - (4)(24)}}{4 + 24} \]
Calculating inside the square root:
\[ \tan \theta = \frac{2\sqrt{25 - 96}}{28} \]
This simplifies to:
\[ \tan \theta = \frac{2\sqrt{-71}}{28} \]
Since the value under the square root is negative, it indicates that the angle is complex, implying that the lines do not intersect in the real plane.

### Final Result
Thus, the angle between the pair of lines is complex, and we can express it in terms of the imaginary unit \(i\):
\[ \tan \theta = \frac{2i\sqrt{71}}{28} = \frac{i\sqrt{71}}{14} \]