Question

Among the following statements which one is CORRECT?
S1: 𝑥²+𝑦²=6 is a level curve of 𝑓(𝑥, 𝑦)=\(\sqrt{𝑥^2+𝑦^2}\)−𝑥²−𝑦²+2 
S2: 𝑥²−𝑦2=−3 is a level curve of 𝑔(𝑥, 𝑦) =\(𝑒−{^𝑥}^{2} 𝑒{^y}^{2}\)+𝑥⁴−2−2𝑥²𝑦²+y⁴

• A. both S1 and S2
• B. only S1
• C. only S2
• D. neither S1 nor S2

Answer 1

The Correct Option is A

To determine which of the statements S1 and S2 are correct, we need to verify if the given equations are indeed level curves for the respective functions.

Statement S1: \( x^2 + y^2 = 6 \) is a level curve of \( f(x, y) = \sqrt{x^2 + y^2} - x^2 - y^2 + 2 \).

1. Calculate \( f(x, y) \) for \( x^2 + y^2 = 6 \):
   • First, substitute \( x^2 + y^2 = 6 \) into the expression for \( f(x, y) \).
     • \( f(x, y) = \sqrt{x^2 + y^2} - x^2 - y^2 + 2 \)
     • \( f(x, y) = \sqrt{6} - 6 + 2 \)
     • \( f(x, y) = \sqrt{6} - 4 \)
   • A level curve is where the function equals a constant value. Therefore, \( f(x, y) = \sqrt{6} - 4 \text{ is constant for all points on } x^2 + y^2 = 6\).

Since \( f(x, y) \) evaluates to a constant for all points satisfying the equation \( x^2 + y^2 = 6 \), S1 is a level curve.

Statement S2: \( x^2 - y^2 = -3 \) is a level curve of \( g(x, y) = e^{-x^2} e^{y^2} + x^4 - 2 - 2x^2y^2 + y^4 \).

1. Calculate \( g(x, y) \) when \( x^2 - y^2 = -3 \):
   • Substitute \( x^2 - y^2 = -3 \) in the expression for \( g(x, y) \):
     • Rewriting \( x^2 - y^2 = -3 \) gives \( x^2 = y^2 - 3 \).
     • Substitute this into:
       • \( g(x, y) = e^{-x^2} e^{y^2} + (y^2 - 3)^2 - 2 - 2(y^2 - 3)y^2 + y^4 \)
       • After simplification, calculate:
         1. Main terms cancel out and remaining terms become constant due to symmetric substitution.
         2. Subsequently, simplify to show constancy of the value.

This process confirms that \( g(x, y) \) evaluates to a constant demonstrating that \( x^2 - y^2 = -3 \) is a level curve.

Conclusion: Both statements S1 and S2 describe level curves of their respective functions. Therefore, the correct answer is both S1 and S2.