Question

If $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$ are two linear equations, and if $\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2}$, then:

• A. Lines are parallel
• B. Lines are coincident
• C. Lines are intersecting
• D. None of these

Hint:

If $\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2}$, then both linear equations represent the same line, i.e., they are coincident.

Answer 1

The Correct Option is B

Step 1: Write the general form of the lines.
The two given linear equations are: \[ a_1x + b_1y + c_1 = 0 \quad \text{and} \quad a_2x + b_2y + c_2 = 0 \]
Step 2: Recall the condition for the nature of lines.
- If $\dfrac{a_1}{a_2} \neq \dfrac{b_1}{b_2}$, the lines intersect.
- If $\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} \neq \dfrac{c_1}{c_2}$, the lines are parallel.
- If $\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2}$, the lines are coincident.

Step 3: Apply the given condition.
Since it is given that \[ \dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2}, \] it means both lines represent the same equation and lie on each other.

Step 4: Conclusion.
Therefore, the two lines are
coincident.
Final Answer
Final Answer: \[ \boxed{\text{Lines are coincident.}} \]