Question

1. the equation $y^2 - x^2 = 1$ represents which conic section? hyperbola ellipse circle parabola

Explanation:

Step1: Recall hyperbola standard form

The standard form of a hyperbola that opens up/down is $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$.

Step2: Match given equation to form

The given equation is $y^2 - x^2 = 1$, which can be rewritten as $\frac{y^2}{1^2} - \frac{x^2}{1^2} = 1$. This fits the hyperbola standard form with $a=1$, $b=1$.

Step3: Eliminate other conic types

• Circle: Has form $x^2+y^2=r^2$ (no subtraction)
• Ellipse: Has form $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ (addition)
• Parabola: Has only one squared variable

Thus, the equation represents a hyperbola.

Answer:

A. hyperbola