Find the equation of a hyperbola with asymptote at x=4 and y=2 that passes through (3,4)?

Question

Find the equation of a hyperbola with asymptote at x=4 and y=2 that passes through (3,4)

Yes i know asymptotes are lines.. thats the question. it was given to me i didnt make it.

Yes it is centred.

Answer 1

Hi,

The equation is y = -2/(x - 4) + 2.

It has a vertical asymptote at x = 4 because that number would make the denominator equal to 0.

There is also a horizontal asymptote at y = 2 because of the constant 2. That's because the fraction keeps getting closer and closer to zero the further out you go on the graph. That is -2/(1004 - 4) = -.002. that means the y value is almost 2.

If you plug 3 in for x in the equation, you will see that y does work out to be 4.

y = -2/(x - 4) + 2
y = -2/(3 - 4) + 2
y = -2/(-1) + 2
y = 2 + 2
y = 4 This shows that (3,4) is on the graph.


I hope that helps!!

Answer 2

Equations of asymptotes are [(x-2)^2]/9 - [(y+a million)^2]/4 = 0 => (x - 2)/3 = (y + a million)/2 and (x -2)/3 = -(y + a million)/2 => y + a million = (2/3)x - 4/3 and -y - a million = (2/3)x - 4/3 => y = (2/3)x - 7/3 and y = -(2/3)x + a million/3 e book answer is right. basically bear in innovations an common formulation for asymptotes of hyperbola (x - x')^2/a^2 - (y - y')^2/b^2 = a million is (x - x')^2/a^2 - (y - y')^2/b^2 = 0 (be conscious the easy step of adjusting a million via 0.)

Answer 3

This is a hyperbola that is tilted 45° in relation to the x and y axes. The center is at the intersection of the asymptotes. The equation then is:

y - 2 = a/(x - 4)

Now we need to solve for a. Plug in the point (3,4).

4 - 2 = a/(3-4) = a/-1 = -a
2 = -a
a = -2

So the equation of the pair of hyperbolas is:

y - 2 = -2/(x - 4)
or
y = -2/(x - 4) + 2

Answer 4

Hmm. Is there any other information? Is it centered at the origin? If it is, then it can be solved.

If it is centered at the origin, then the equation can be written as:

y^2/(q)^2-x^2/(2q)^2=1

if y=3 and x=4, then:

1=9/(q)^2-16/(2q)^2

q^2=9-4
q=sqrt(5)

so the equation is 1=y^2/5-x^2/20