Question

The joint equation of the pair of lines passing through the point of intersection of the lines \[ 2x^2 - xy - 15y^2 - 7x + 32y - 9 = 0 \] and parallel to the coordinate axes is

• A. \( xy - x - 2y + 2 = 0 \)
• B. \( xy + x + 2y - 2 = 0 \)
• C. \( xy + x + 2y + 2 = 0 \)
• D. \( xy - x - 2y - 2 = 0 \)

Hint:

For lines parallel to coordinate axes passing through a point \((a,b)\), directly use \((x-a)(y-b)=0\).

Answer 1

The Correct Option is A

Step 1: Understand the requirement.
Lines parallel to the coordinate axes are of the form \[ x = a \quad \text{and} \quad y = b \] Their joint equation is \[ (x-a)(y-b)=0 \Rightarrow xy - bx - ay + ab = 0 \]

Step 2: Find the point of intersection of the given lines.
The given second-degree equation represents a pair of straight lines intersecting at a point \( (a,b) \). By comparing with the standard form and evaluating the intersection point, we obtain \[ (a,b) = (2,1) \]

Step 3: Form the joint equation.
\[ (x-2)(y-1)=0 \] \[ xy - x - 2y + 2 = 0 \]

Step 4: Conclusion.
Hence, the required joint equation is \[ \boxed{xy - x - 2y + 2 = 0} \]