Question

find the standard form of the equation of the hyperbola shown.
the standard form of the equation is

Explanation:

Step1: Identify hyperbola type and center

Horizontal transverse axis, center at (0,0). Standard form: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$.

Step2: Find a from vertices

Vertices at (±6,0), so $a=6$, $a^2=36$.

Step3: Find b using asymptote slope

Asymptote slope $\frac{b}{a} = \frac{2}{3}$. $\frac{b}{6} = \frac{2}{3} \Rightarrow b=4$, $b^2=16$.

Step4: Write standard equation

Substitute $a^2=36$ and $b^2=16$ into the form.

Answer:

$\frac{x^2}{36} - \frac{y^2}{16} = 1$