Relation of Parallel Lines on XY-Plane

📺 Video Explanation

📝 Question

Let \( L \) be the set of all lines in the XY-plane. Define relation \( R \) as:

\[ R = \{(L_1, L_2) : L_1 \text{ is parallel to } L_2\} \]

Show that \( R \) is an equivalence relation. Also find all lines related to the line \( y = 2x + 4 \).

---

✅ Solution

🔹 Step 1: Reflexive

Every line is parallel to itself: \[ L \parallel L \]

So, \[ (L, L) \in R \]

✔ Therefore, the relation is Reflexive. :contentReference[oaicite:0]{index=0}

---

🔹 Step 2: Symmetric

If one line is parallel to another: \[ L_1 \parallel L_2 \]

Then: \[ L_2 \parallel L_1 \]

So, \[ (L_2, L_1) \in R \]

✔ Therefore, the relation is Symmetric. :contentReference[oaicite:1]{index=1}

---

🔹 Step 3: Transitive

If: \[ L_1 \parallel L_2 \text{ and } L_2 \parallel L_3 \]

Then: \[ L_1 \parallel L_3 \]

So, \[ (L_1, L_3) \in R \]

✔ Therefore, the relation is Transitive. :contentReference[oaicite:2]{index=2}

---

🎯 Final Conclusion

✔ Reflexive: Yes
✔ Symmetric: Yes
✔ Transitive: Yes

\[ \therefore R \text{ is an equivalence relation} \]

---

🔹 Lines Related to \( y = 2x + 4 \)

All lines parallel to a given line have the same slope.

Slope of \( y = 2x + 4 \) is: \[ m = 2 \]

So, all lines of the form: \[ y = 2x + c \quad (c \in \mathbb{R}) \] are parallel to \( y = 2x + 4 \).

Thus, the set of all lines related to \( y = 2x + 4 \) is:

\[ \{\, y = 2x + c \;|\; c \in \mathbb{R} \,\} \]

---

🚀 Exam Insight

• Parallel lines ⇒ same slope
• Each slope defines one equivalence class
• Equivalence classes = family of parallel lines