Question

Geometry
Which is the equation of a hyperbola with directrices at x = ±2 and foci at (4, 0) and (−4, 0)?

Answer 1

@dUdE

Answer 2

I despise conic sections and I never enjoyed learning about it but I just brushed up on it since it seems like no one else wants to bite.
I found this website to be particularly helpful in refreshing my knowledge and would advise you to take a look at it too. https://courses.lumenlearning.com/waymakercollegealgebra/chapter/equations-of-hyperbolas/

When the coordinates of the foci are \( (\pm c, ~0)\), the standard form of the equation of a hyperbola with center (0, 0) is

\( \dfrac{{x}^{2}}{{a}^{2}}-\dfrac{{y}^{2}}{{b}^{2}}=1\)

When the coordinates of the foci are \( (0,~ \pm c)\), the standard form of the equation of a hyperbola with center (0, 0) is
\(\dfrac{{y}^{2}}{{a}^{2}}-\dfrac{{x}^{2}}{{b}^{2}}=1\)


We're given the foci as (-4, 0) and (4, 0). Which standard form should we use?
This should help you eliminate two answer choices almost immediately.

Answer 3

Now, to finally narrow it down to the correct answer.

The equation of directrices are
\( x = \pm\dfrac{a^2}{c}\)

We're given that the hyperbola has a directrices at \(x = \pm 2\)

We know what c is because that's our foci. It's in the form \( ( \pm c, ~ 0)\) so our c is is 4.

To find out what `a` and `b` is and get our equation, we need to first solve for `a` using \( x = \pm\dfrac{a^2}{c}\)

since we know \(x = \pm 2\), we can set it up as
\(\dfrac{a^2}{c} = 2\)

plug in c = 4 and solve for a!

To solve for b, you'll need to use the formula
\( c^2 = a^2 + b^2\)

Answer 4

Let me know what you get or if you get stuck anywhere along the way!

Answer 5

i need help

Answer 6

g-guys

Answer 7

\(\color{#0cbb34}{\text{Originally Posted by}}\) @AZ
Let me know what you get or if you get stuck anywhere along the way!
\(\color{#0cbb34}{\text{End of Quote}}\)
wowwww thankz ur such a genius

Answer 8

It was my pleasure!