Two straight lines pass through the origin (𝑥, 𝑦) = (0,0). One of them passes
through the point (𝑥ଵ, 𝑦ଵ) = (1,3) and the other passes through the point
(𝑥ଶ, 𝑦ଶ) = (1,2).
What is the area enclosed between the straight lines in the interval [0, 1] on
the 𝑥-axis?
Options:
(A) Q.5
(B) 1.0
(C) 1.5
(D) 2.0
Official Answer Key:
A
Step-by-Step Solution & Explanation
### **Official Solution & Concept Analysis (GATE BT 2022 Q.2)**
**1. Governing Concept**:
- **Topic**: Coordinate Geometry & Integral Calculus (Area Between Curves)
- Line 1 passes through $(0,0)$ and $(1,3)$:
$y_1 = \frac{3 - 0}{1 - 0} x = 3x$
- Line 2 passes through $(0,0)$ and $(1,2)$:
$y_2 = \frac{2 - 0}{1 - 0} x = 2x$
- The area enclosed between the two straight lines in the interval $x \in [0, 1]$:
$\text{Area} = \int_0^1 (y_1 - y_2) \, dx = \int_0^1 (3x - 2x) \, dx = \int_0^1 x \, dx = \left[ \frac{x^2}{2} \right]_0^1 = \frac{1}{2} = 0.5$
**2. Option Analysis**:
- **Option (A) [0.5] is CORRECT**: Directly evaluates to $0.5$ square units.
- **Option (B) [1.0] is INCORRECT**: Evaluates to $1.0$ if forgetting to divide by 2.
- **Option (C) [1.5] is INCORRECT**: Area under line 2 alone.
- **Option (D) [2.0] is INCORRECT**: Area of triangle formed by $(1,2)$ and $(1,0)$ or similar.
**3. Final Answer**:
- **Correct Choice**: **A**
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