The combined equation of the lines through origin and perpendicular to the pair of lines `3x^(2) + 4xy - 5y^(2) = 0` is…

Options

A. `5x^(2) + 4xy - 3y^(2) = 0`
B. `3x^(2) + 4xy - 5y^(2) = 0`
C. `3x^(2) - 4xy + 5y^(2) = 0`
D. `5x^(2) + 4xy + 3y^(2) = 0`

Answer

To find the combined equation of the lines through the origin and perpendicular to the pair of lines given by the equation \(3x^2 + 4xy - 5y^2 = 0\), we can follow these steps:

### Step 1: Identify the coefficients
The given equation of the pair of lines is in the form \(Ax^2 + 2Hxy + By^2 = 0\). Here, we have:
- \(A = 3\)
- \(H = 4/2 = 2\)
- \(B = -5\)

### Step 2: Calculate the slopes of the lines
The slopes \(m_1\) and \(m_2\) of the lines can be found using the relationships:
- Sum of the slopes:
\[ m_1 + m_2 = -\frac{2H}{A} = -\frac{2 \times 2}{3} = -\frac{4}{3} \]
- Product of the slopes:
\[ m_1 \cdot m_2 = \frac{A}{B} = \frac{3}{-5} = -\frac{3}{5} \]

### Step 3: Set up the equations
We can set up the equations based on the sum and product of the slopes:
1. \(m_1 + m_2 = -\frac{4}{3}\)
2. \(m_1 m_2 = -\frac{3}{5}\)

### Step 4: Solve for the slopes
Let \(m_1\) and \(m_2\) be the roots of the quadratic equation:
\[ x^2 - (m_1 + m_2)x + m_1 m_2 = 0 \]
Substituting the values we found:
\[ x^2 + \frac{4}{3}x - \frac{3}{5} = 0 \]
To eliminate the fractions, we can multiply through by 15 (the least common multiple of 3 and 5):
\[ 15x^2 + 20x - 9 = 0 \]

### Step 5: Use the quadratic formula
Now we can use the quadratic formula to find the slopes:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \(a = 15\), \(b = 20\), and \(c = -9\):
\[ x = \frac{-20 \pm \sqrt{20^2 - 4 \cdot 15 \cdot (-9)}}{2 \cdot 15} \]
Calculating the discriminant:
\[ 20^2 = 400 \]
\[ -4 \cdot 15 \cdot (-9) = 540 \]
\[ \text{Discriminant} = 400 + 540 = 940 \]
Now substituting back:
\[ x = \frac{-20 \pm \sqrt{940}}{30} \]
\[ \sqrt{940} = \sqrt{4 \cdot 235} = 2\sqrt{235} \]
Thus,
\[ x = \frac{-20 \pm 2\sqrt{235}}{30} = \frac{-10 \pm \sqrt{235}}{15} \]

### Step 6: Find the slopes
The slopes are:
\[ m_1 = \frac{-10 + \sqrt{235}}{15}, \quad m_2 = \frac{-10 - \sqrt{235}}{15} \]

### Step 7: Find the slopes of the perpendicular lines
The slopes of the lines perpendicular to these slopes are given by:
\[ m_3 = -\frac{1}{m_1}, \quad m_4 = -\frac{1}{m_2} \]

### Step 8: Write the combined equation
The combined equation of the lines through the origin with slopes \(m_3\) and \(m_4\) is:
\[ y^2 + (m_3 + m_4)xy + m_3m_4x^2 = 0 \]

### Step 9: Substitute the values
Using the values for \(m_3\) and \(m_4\) and substituting them into the equation will yield the final combined equation.