Question

The perpendicular distance between the lines given by \( (x - 2y + 1)^2 + \text{k(x - 2y + 1) = 0 \) is \( \sqrt{5} \), then k =}

• A. 5
• B. 2
• C. 4
• D. 6

Hint:

Equations of the form \( f(x,y)^2 + k \cdot f(x,y) = 0 \) always represent two parallel lines.

Answer 1

The Correct Option is A

Step 1: Concept Factor the equation to find two individual parallel lines and use the distance formula \( d = \frac{|c_1-c_2|}{\sqrt{a^2+b^2}} \).

Step 2: Meaning Factor: \( (x - 2y + 1) [ (x - 2y + 1) + k ] = 0 \). Line 1: \( x - 2y + 1 = 0 \). Line 2: \( x - 2y + (1 + k) = 0 \).

Step 3: Analysis Distance = \( \frac{|(1+k) - 1|}{\sqrt{1^2+(-2)^2}} = \frac{|k|}{\sqrt{5}} \). Given distance is \( \sqrt{5} \), so \( \frac{|k|}{\sqrt{5}} = \sqrt{5} \implies |k| = 5 \).

Step 4: Conclusion The value of \( k \) is 5. Final Answer: (A)