Question

If the pair of linear equations : \( a_1x + b_1y + c_1 = 0 \) and \( a_2x + b_2y + c_2 = 0 \) is consistent and dependent, then

• A. \( \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \)
• B. \( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \)
• C. \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2} \)
• D. \( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \)

Hint:

Dependent lines are just the same line written differently. For example, \( x+y=2 \) and \( 2x+2y=4 \).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
A system of equations is 'consistent' if it has at least one solution. It is 'dependent' if it has infinitely many solutions (the two lines coincide).
Step 2: Detailed Explanation:
The three conditions for linear equations are:
1. Unique Solution: \( \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \) (Consistent and Independent).
2. No Solution: \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2} \) (Inconsistent).
3. Infinitely Many Solutions: \( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \) (Consistent and Dependent).
The question explicitly asks for the consistent and dependent case.
Step 3: Final Answer:
The condition is \( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \).