Question

The pair of linear equations $x-y-1=0$ and $x-2y+2=0$ represents _____ lines.

• A. Coinciding
• B. Intersecting
• C. Parallel
• D. Curved

Hint:

To identify the nature of two lines quickly: \[ \frac{a_1}{a_2}\neq \frac{b_1}{b_2} \Rightarrow \text{Intersecting} \] \[ \frac{a_1}{a_2}= \frac{b_1}{b_2}\neq \frac{c_1}{c_2} \Rightarrow \text{Parallel} \] \[ \frac{a_1}{a_2}= \frac{b_1}{b_2}= \frac{c_1}{c_2} \Rightarrow \text{Coincident} \]

Answer 1

The Correct Option is B

Concept: For two linear equations: \[ a_1x+b_1y+c_1=0 \] and \[ a_2x+b_2y+c_2=0 \] their nature depends on the ratios: \[ \frac{a_1}{a_2}, \quad \frac{b_1}{b_2}, \quad \frac{c_1}{c_2} \] Rules:
• If \[ \frac{a_1}{a_2}\neq \frac{b_1}{b_2} \] then the lines intersect at one point.
• If \[ \frac{a_1}{a_2}=\frac{b_1}{b_2}\neq \frac{c_1}{c_2} \] then the lines are parallel.
• If \[ \frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2} \] then the lines coincide.

Step 1: Write the equations clearly.
Given equations: \[ x-y-1=0 \] \[ x-2y+2=0 \] Comparing with standard form: \[ a_1=1,\quad b_1=-1,\quad c_1=-1 \] \[ a_2=1,\quad b_2=-2,\quad c_2=2 \]

Step 2: Find the ratios.
First ratio: \[ \frac{a_1}{a_2}=\frac{1}{1}=1 \] Second ratio: \[ \frac{b_1}{b_2}=\frac{-1}{-2}=\frac{1}{2} \]

Step 3: Compare the ratios.
We observe: \[ 1 \neq \frac{1}{2} \] Thus, \[ \frac{a_1}{a_2}\neq \frac{b_1}{b_2} \] Hence, the two lines intersect each other at exactly one point. Therefore, the equations represent: \[ \boxed{\text{Intersecting lines}} \]