Question

Which is the equation of a hyperbola with directrices at x = ±2 and foci at (6, 0) and (−6, 0)?

Answer 1

Answer:

$\displaystyle \frac{x^2}{12}-\frac{y^2}{24}=1$

Step-by-step explanation:

Since the directrices are vertical lines:

• Use the hyperbola equation $\displaystyle \frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1$
• Foci are $(\pm c,0)$
• $a^2+b^2=c^2$
• $a$ and $b$ are half the lengths of the tranverse and conjugate axes respectively
• Directrices are located at the lines $\displaystyle x=\pm\frac{a^2}{c}$
• We assume the center to be $(h,k)\rightarrow(0,0)$

Since we already know our directrices are at the lines $x=\pm2$ and we know that $c=\pm6$ from our foci, we can solve for the value of $a^2$:

$x=\frac{a^2}{c}\\\\2=\frac{a^2}{6}\\\\12=a^2$

This allows us to solve for the value of $b^2$:

$a^2+b^2=c^2\\\\12+b^2=6^2\\\\12+b^2=36\\\\b^2=24$

Thus, our equation for the hyperbola is $\displaystyle \frac{x^2}{12}-\frac{y^2}{24}=1$.