Question

Which is the equation of a hyperbola with directrices at x = ±2 and foci at (5, 0) and (−5, 0)?

Answer 1

The required equation of the hyperbola is expressed as

$\frac{x^2}{10}- \frac{y^2}{15}=1$

The standard form for calculating the equation of a parabola along the x-axis is expressed as:

$\frac{(x-k)^2}{a^2}-\frac{(y-h)^2}{b^2} = 1 \ ............. 1$ where:

(h, k) is the center

(k±c, h) are the foci

$x=h\pm\frac{a^2}{c}$ is the directrix

From the question given, we can see that foci = (±5, 0)

k±c, h = ±5, 0

k = 0

h = 0

c = 5

From the directrix,

$x=h\pm\frac{a^2}{c}\\x =0\pm\frac{a^2}{5}\\\pm2 = \pm\frac{a^2}{5}\\a^2 = 10$

Also, we need to know that;

a²+b² = c²

10 + b² = 5²

b² = 25 - 10

b² = 15

Substituting the gotten values into the equation of a hyperbola;

$\frac{(x-0)^2}{10}- \frac{(y-0)^2}{15} =1\\\frac{x^2}{10}- \frac{y^2}{15} =1$

This gives the required equation of the hyperbola

Learn more on the equation of hyperbola here: brainly.com/question/20409089