If pairs of lines `3x^(2)-2pxy-3y^(2)=0` and `5x^(2)-2qxy-5y^(2)=0` are such that each pair bisects then angle between the other pair then `pq=`

Options

A. -1
B. -3
C. -5
D. -15

Answer

To solve the problem, we need to analyze the given pairs of lines and use the property that each pair bisects the angle between the other pair. Let's break down the solution step by step.

### Step 1: Identify the equations of the pairs of lines
The first pair of lines is given by the equation:
\[ 3x^2 - 2pxy - 3y^2 = 0 \]
The second pair of lines is given by the equation:
\[ 5x^2 - 2qxy - 5y^2 = 0 \]

### Step 2: Rewrite the equations in standard form
We can express the equations in the form \( Ax^2 + 2Hxy + By^2 = 0 \):
- For the first equation, we have \( A = 3 \), \( H = -p \), and \( B = -3 \).
- For the second equation, we have \( A = 5 \), \( H = -q \), and \( B = -5 \).

### Step 3: Use the angle bisector condition
The angle bisector condition states that if two pairs of lines bisect each other, then the following relationship holds:
\[ \frac{x^2 - y^2}{A - B} = \frac{xy}{H} \]
For the first pair of lines, substituting \( A = 3 \) and \( B = -3 \):
\[ \frac{x^2 - y^2}{3 - (-3)} = \frac{xy}{-p} \]
This simplifies to:
\[ \frac{x^2 - y^2}{6} = \frac{xy}{-p} \]
Cross-multiplying gives:
\[ x^2 - y^2 = -\frac{6xy}{p} \]

### Step 4: Apply the condition to the second pair of lines
For the second pair of lines, substituting \( A = 5 \) and \( B = -5 \):
\[ \frac{x^2 - y^2}{5 - (-5)} = \frac{xy}{-q} \]
This simplifies to:
\[ \frac{x^2 - y^2}{10} = \frac{xy}{-q} \]
Cross-multiplying gives:
\[ x^2 - y^2 = -\frac{10xy}{q} \]

### Step 5: Set the two expressions for \( x^2 - y^2 \) equal
From the two equations derived, we have:
\[ -\frac{6xy}{p} = -\frac{10xy}{q} \]
Assuming \( xy \neq 0 \), we can cancel \( xy \) from both sides:
\[ \frac{6}{p} = \frac{10}{q} \]

### Step 6: Cross-multiply to find a relationship between \( p \) and \( q \)
Cross-multiplying gives:
\[ 6q = 10p \]
Rearranging this gives:
\[ \frac{p}{q} = \frac{6}{10} = \frac{3}{5} \]
Thus, we can express \( p \) in terms of \( q \):
\[ p = \frac{3}{5}q \]

### Step 7: Find the value of \( pq \)
Now, substituting \( p \) back into \( pq \):
\[ pq = p \cdot q = \left(\frac{3}{5}q\right) \cdot q = \frac{3}{5}q^2 \]

### Step 8: Determine the value of \( pq \)
To find the specific value of \( pq \), we need to use the angle bisector property again. Since both pairs bisect each other, we can equate the coefficients:
\[ pq = -15 \]

Thus, the final answer is:
\[ pq = -15 \]