The line $\frac{x - 2}{3} = \frac{y + 1}{2} = \frac{z - 1}{-1}$ intersects the curve $xy = c^2, z = 0$ if $c$ is equal to:
Step-by-Step Solution
Key Concept: Setting $z = 0$ in the parametric line equations determines the intersection points, which then satisfy the hyperbolic curve equation.
For points where the line intersects the curve with $z = 0$, we use parametric form $\frac{x-2}{3} = \frac{y+1}{2} = \frac{0-1}{-1} = 1$, giving $x = 5$ and $y = 1$. Substituting into $xy = c^2$ yields $c = \pm\sqrt{5}$.
Correct Answer: Let me work through this step-by-step.
**Finding the intersection point:**
The line is given in symmetric form: $\frac{x - 2}{3} = \frac{y + 1}{2} = \frac{z - 1}{-1}$
For intersection with the curve $xy = c^2, z =