Biomolecules, enzyme kinetics

### **Step-by-Step Solution & Mathematical Derivation (GATE BT 2023 Q.35)** **1. Governing Concept & Definition**: - **Topic**: Coordinate Geometry & Intersection of Curves - The distance $d$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ in the Cartesian plane is: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ **2. Detailed Step-by-Step Calculation**: - **Given Equations**: 1. Parabola: $x^2 + y = 7 \implies y = 7 - x^2$ 2. Line: $x + y = 7 \implies y = 7 - x$ - **Find Intersection Points**: $7 - x^2 = 7 - x \implies x^2 - x = 0 \implies x(x - 1) = 0$ - At $x_1 = 0$: $y_1 = 7 - 0 = 7 \implies \text{Point } P_1 = (0, 7)$ - At $x_2 = 1$: $y_2 = 7 - 1 = 6 \implies \text{Point } P_2 = (1, 6)$ - **Calculate Distance**: $d = \sqrt{(1 - 0)^2 + (6 - 7)^2} = \sqrt{1^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2} \approx 1.4142 \approx 1.41$ **3. Final Answer**: - **Calculated Value**: **1.41** - **Official GATE Accepted Range**: **1.40 to 1.42**