Question

Help Please

A hyperbola centered at the origin has a vertex at (−6, 0) and a focus at (10, 0). Which are the equations of the directrices?

A) x= +or- 3/5
B) y= +or- 3/5
C) x= +or- 18/5
D) y= +or- 18/5

Answer 1

I believe the answer is c. hope this helps

Answer 2

If a hyperbola is centered at the origin and has vertex and focus on the x-axis, then its equation is

`(x^2)/(a^2)-(y^2)/(b^2)=1.`

1. The vertex A of hyperbola has coordinates (-6,0), then the distance from the origin (center of the hyperbola) to the vertex is

`a=6.`

2. The focus F has coordinates (10,0), then the distance from the origin to the focus is

`c=10.`

3. Find b, using formula
`c^2=a^2+b^2:`

`10^2=6^2+b^2,\\ \\b^2=100-36=64,\\ \\b=8.`

4. The equation of the hyperbola is

`(x^2)/(36)-(y^2)/(64)=1.`

5. The directrices' equations are

`x=-(a)/(e),\ x=(a)/(e),`

where
`e=(c)/(a).`

In your case,

`e=(10)/(6)`

and directrices' equations are

`x=-(6)/((10)/(6))=-3.6,\ x=(6)/((10)/(6))=3.6.`

Answer: x=-3.6, x=3.6, correct choice is C