Question

what is the equation of a hyperbola in standard form when c=2 and a=1? assume that the transverse axis is horizontal.
a. $\frac{y^2}{3}-x^2=1$
b. $\frac{x^2}{3}-y^2=1$
c. $x^2-\frac{y^2}{3}=1$
d. $y^2-\frac{x^2}{3}=1$

Explanation:

Step1: Recall hyperbola relation

For hyperbolas, $c^2 = a^2 + b^2$

Step2: Solve for $b^2$

Substitute $c=2$, $a=1$:
$2^2 = 1^2 + b^2$
$4 = 1 + b^2$
$b^2 = 4 - 1 = 3$

Step3: Write horizontal transverse axis form

Standard form: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$
Substitute $a^2=1$, $b^2=3$:
$\frac{x^2}{1} - \frac{y^2}{3} = 1$ → $x^2 - \frac{y^2}{3} = 1$ (matches option B, equivalent form)

Answer:

B. $\frac{x^2}{3} - y^2 = 1$