Subtopic: Fundamentals of Biomolecules, enzyme kinetics
Question Statement:
A straight line y = x - 1 intersects a circle with center at x = 1, y = 1 and radius
of magnitude 1 at two points. The length of the chord formed by this intersection is
_______. (rounded off to three decimal places)
Official Answer Key:
1.414 to 1.414
Step-by-Step Solution & Explanation
### **Step-by-Step Solution & Mathematical Derivation (GATE BT 2026 Q.53)**
**1. Governing Concept & Definition**:
- **Topic**: Engineering Mathematics & Geometry of Chords
- The length of a chord intercepted by a circle of radius $R$ at distance $d$ from the center is:
$\text{Length} = 2 \sqrt{R^2 - d^2}$
**2. Detailed Step-by-Step Calculation**:
- **Given**:
- Circle center: $(x_0, y_0) = (1, 1)$, Radius $R = 1$
- Line: $y = x - 1 \implies x - y - 1 = 0$
- **Step 1: Perpendicular distance ($d$) from center $(1, 1)$ to line**:
$d = \frac{|1 - 1 - 1|}{\sqrt{1^2 + (-1)^2}} = \frac{|-1|}{\sqrt{2}} = \frac{1}{\sqrt{2}}$
- **Step 2: Chord Length**:
$\text{Length} = 2 \sqrt{R^2 - d^2} = 2 \sqrt{1^2 - \left(\frac{1}{\sqrt{2}}\right)^2} = 2 \sqrt{1 - \frac{1}{2}} = 2 \sqrt{\frac{1}{2}} = \frac{2}{\sqrt{2}} = \sqrt{2} \approx 1.414$
**3. Final Answer**:
- **Calculated Value**: **1.414**
- **Official GATE Accepted Range**: **1.414 to 1.414**
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