anonymous
  • anonymous
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
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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jamiebookeater
  • jamiebookeater
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jdoe0001
  • jdoe0001
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
jdoe0001
  • jdoe0001
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
jdoe0001
  • jdoe0001
\(\large \pm\sqrt{a^2+b^2}\) 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 \(\pm\sqrt{41}\)

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jdoe0001
  • jdoe0001
still confused?
anonymous
  • anonymous
Somewhat, I think the answer is B going by what you're saying, is that correct?
jdoe0001
  • jdoe0001
dunno, what do you think about "c"?
jdoe0001
  • jdoe0001
or "d"
jdoe0001
  • jdoe0001
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
jdoe0001
  • jdoe0001
if it's moving horizontally, is opening towards the "x" axis, so the hyperbola will have a POSITIVE fraction with it in it
jdoe0001
  • jdoe0001
so, which do you think you could rule out of the options?
anonymous
  • anonymous
So C and D aren't correct?
jdoe0001
  • jdoe0001
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
anonymous
  • anonymous
I still think it's B
jdoe0001
  • jdoe0001
\(\large \pm\sqrt{a^2+b^2}\) is the distance from (0,0) to the vertices now, one of the focus is \(\sqrt{41}\) meaning \(\large \pm\sqrt{a^2+b^2} = \sqrt{41}\)
jdoe0001
  • jdoe0001
well, let's check B
jdoe0001
  • jdoe0001
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 \(\large \sqrt{25+66}\)
jdoe0001
  • jdoe0001
so, does that give you \(\sqrt{41} \large ?\)

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