Question

If the pair of straight lines \(xy - x + y - 1 = 0\) and the line \(x + ky - 3 = 0\) are concurrent, then the value of \(k\) is equal to

• A. \(4\)
• B. \(3\)
• C. \(-1\)
• D. \(2\)

Hint:

For concurrency with a pair of lines, first factor the pair, find their intersection point, and then substitute into the third line.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Concurrency means all three lines intersect at a common point.
Step 2: Key Formula or Approach:
Factorize the pair of lines: \(xy - x + y - 1 = (x+1)(y-1) = 0\).
Lines are \(x = -1\) and \(y = 1\).
Step 3: Detailed Explanation:
The intersection of \(x = -1\) and \(y = 1\) is \((-1, 1)\).
Since the third line is concurrent, it must pass through \((-1, 1)\).
Substituting into \(x + ky - 3 = 0\):
\(-1 + k(1) - 3 = 0 \implies k - 4 = 0 \implies k = 4\).
Step 4: Final Answer:
Value of \(k\) is \(4\).