Subtopic: Fundamentals of Biomolecules, enzyme kinetics
Question Statement:
If the straight lines given by the following two equations are parallel to each other,
the value of a is _____. (answer in integer)
4x + 7y = 6; 3𝒂x + 42y = 24
Official Answer Key:
8 to 8
Step-by-Step Solution & Explanation
### **Step-by-Step Solution & Mathematical Derivation (GATE BT 2026 Q.52)**
**1. Governing Concept & Definition**:
- **Topic**: Engineering Mathematics & Coordinate Geometry
- Two lines $A_1 x + B_1 y + C_1 = 0$ and $A_2 x + B_2 y + C_2 = 0$ are parallel if and only if their slopes are equal:
$m_1 = m_2 \implies -\frac{A_1}{B_1} = -\frac{A_2}{B_2} \implies \frac{A_1}{A_2} = \frac{B_1}{B_2}$
**2. Detailed Step-by-Step Calculation**:
- **Given Lines**:
1. $4x + 7y = 6$
2. $3ax + 42y = 24$
- **Parallel Condition**:
$\frac{4}{3a} = \frac{7}{42}$
$\frac{7}{42} = \frac{1}{6}$
$\frac{4}{3a} = \frac{1}{6} \implies 3a = 24 \implies a = 8$
**3. Final Answer**:
- **Calculated Value**: **8**
- **Official GATE Accepted Range**: **8 to 8**
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