Question

The locus of points(x,y) in the plane satisfying sin2x + sin2y =1 consists of

• A. a circle centered at origin
• B. infinitely many circles that are all centered at the origin
• C. infinitely many lines with slope ±1
• D. finitely many lines with slope ±1

Answer 1

Answer: (C)

infinitely many lines with slope ±1

Answer 2

The given equation is x2 + 2xsin(xy) + 1 = 0.
Solving for sin(xy), we find that it is equal to $\frac{-2x}{(1+x^2)}$, which can also be expressed as $\frac{-2}{(x+1)}$.
Since (x+1) is always greater than or equal to 2, it follows that sin(xy) is less than or equal to -1. This situation is possible only when sin(θ) equals -1 for some angle θ.
Therefore, we have sin(xy) = -1, which implies that xy =$\frac{-π}{2}$.
This equation describes a hyperbola because it is in the form of xy = a constant (in this case, a negative constant, $\frac{-π}{2}$).
So, the correct answer is option (C): infinitely many lines with slope ±1.