Question

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

Answer 1

Answer:

$\frac{x^{2}}{7.198}-\frac{y^{2}}{28.794} = 1$

Step-by-step explanation:

The formula of the directrices are:

$y = \pm 2 \cdot x$

A relationship between the hyperbola constants is derived from the absolute value of the slope of the directrix:

$\frac{b}{a} = 2$

The location of the foci indicates that hyperbola is vertical and centered at origin. The distance between foci is equal to $2\cdot c$. Then, the vertical parameter is:

$2\cdot c = 12$

$c = 6$

One of the hyperbola coefficients is determined with the help of the Pythagorean Theorem:

$c = \sqrt{a^{2}+b^{2}}$

$c = \sqrt{a^{2}+4\cdot a^{2}}$

$c = \sqrt{5\cdot a^{2}}$

$c = \sqrt{5}\cdot a$

$a = \frac{\sqrt{5}\cdot c}{5}$

$a \approx 2.683$

The remaining parameter is found by means of the relationship described previously:

$b = 2\cdot a$

$b = 5.366$

Lastly, the equation of the hyperbola in standard form is:

$\frac{x^{2}}{7.198}-\frac{y^{2}}{28.794} = 1$