Question

Completely lost :(

Which of the following is the hyperbola that is centered at the origin with one focus sqrt 41,0 and one vertex (5,0)


A: x^2/25-y^2/16=1
B: x^2/25-y^2/66=1
C:y^2/16-x^2/25=1
D:y^2/66-x^2/25=1

Answer 1

well, on a hyperbola, the fraction with the POSITIVE sign, is where the hyperbola is opening up, or moving about,
so, since it's center is at the origin, (0,0) and a vertex is at (5,0) it means is moving HORIZONTALLY,
so that'd rule out 2 of those guys

Answer 2

now, it has a focus at \(+\sqrt{41}\), the other focus will be at \(-\sqrt{41}\)

so, since you already ruled out 2 of those, then, now just need to test the other 2 remaining ones, which one will give you a \(\pm\sqrt{41} \) for the focus formula

the hyperbola focus formula will be
$$ \large
\pm\sqrt{a^2+b^2}
$$
in this case your a = 25 for both
and b = 16 and 66

Answer 3

\(\large
\pm\sqrt{a^2+b^2}\) is the distance from the center of the hyperbola, to the focus, in this case, the center is (0,0) so the distance will be just \(\pm\sqrt{41}\)

Answer 4

still confused?

Answer 5

Somewhat, I think the answer is B going by what you're saying, is that correct?

Answer 6

dunno, what do you think about "c"?

Answer 7

or "d"

Answer 8

well, on a hyperbola, the fraction with the POSITIVE sign, is where the hyperbola is opening up, or moving about,
so, since it's center is at the origin, (0,0) and a vertex is at (5,0) it means is moving HORIZONTALLY,

there are only 2 up there, where fraction with the "x" in it has a positive sign

Answer 9

if it's moving horizontally, is opening towards the "x" axis, so the hyperbola will have a POSITIVE fraction with it in it

Answer 10

so, which do you think you could rule out of the options?

Answer 11

So C and D aren't correct?

Answer 12

right, c and d, are have a positive fraction with "y" in it, so they're opening upwards/downwards, so, c and d are out

Answer 13

I still think it's B

Answer 14

\(\large \pm\sqrt{a^2+b^2}\) is the distance from (0,0) to the vertices
now, one of the focus is \(\sqrt{41}\)
meaning \(\large \pm\sqrt{a^2+b^2} = \sqrt{41}\)

Answer 15

well, let's check B

Answer 16

in B you have \(a^2 = 25 \implies a = 5\)
you also have \(b^2 = 66 \implies b = \sqrt{66}\)

so the focus will be at \(\large \sqrt{25+66}\)

Answer 17

so, does that give you \(\sqrt{41} \large ?\)