Question

The joint equation of a pair of lines passing through \((2,3)\) and parallel to the lines \[ x^2 - y^2 = 0 \] is

• A. \(x^2 - y^2 - 4x + 6y - 5 = 0\)
• B. \(x^2 - y^2 - 4x + 6y = 0\)
• C. \(x^2 - y^2 - 4x + 6y + 17 = 0\)
• D. \(x^2 - y^2 - 4x + 6y + 2 = 0\)

Hint:

For joint equations of lines parallel to a given pair, keep the quadratic part unchanged and determine constants using the given point.

Answer 1

The Correct Option is A

Step 1: Interpret the given equation.
The equation \[ x^2 - y^2 = 0 \] represents a pair of straight lines given by
\[ (x - y)(x + y) = 0 \] which are the lines \(y = x\) and \(y = -x\).
Step 2: Write the general form of parallel lines.
A pair of lines parallel to \(x^2 - y^2 = 0\) is given by
\[ x^2 - y^2 + 2gx + 2fy + c = 0 \] Step 3: Use the point condition.
Since the lines pass through \((2,3)\), substitute \(x = 2,\; y = 3\):
\[ 4 - 9 + 4g + 6f + c = 0 \Rightarrow -5 + 4g + 6f + c = 0 \] Step 4: Obtain the required equation.
Comparing with the correct option, the required joint equation is
\[ x^2 - y^2 - 4x + 6y - 5 = 0 \]