writing equation of hyperbola given foci and asymptote?

[poisonfartdrogs]

I have always had trouble grasping proof behind why algebraic problems work, and it's resulted in me being somewhat unable to think flexibly when trying to find certain variables. The question I need help understanding the process of solving is: Find the equation of the hyperbola given the following: foci (0, +or-8) and asymptotes y=+or-1/2x

I looked in the back of the book, and the solution is 5y^2/64 - 5x^2/256 = 1, but I can't for the life of me figure out how to get to that solution. I know that c=+or-8 and that the asymptotes are of the form y=+or-(a/b)x. My teacher told me because the asymptote gives a & b in terms of y=+or-(a/b)x, I should view 1 & 2 as a & b respectively. My teacher then said that because a/b=1/2 in this case, I needed to change it to solve for either a or b, and he wrote down a=1/2b and told me I will need to substitute that somewhere to proceed finding the equation for the hyperbola, but not only do I not know where to substitute it, I do not know how he solved it that way or why. If 1=a, and 2=b, how did he algebraically manipulate it to read a=1/2b? It seems like he added variables a and b into an equation that already had numbers to stand for variables a & b.

I hope I articulated my confusion well enough. Once again, I listed the answer to this equation according to the back of my book, as well as what the original problem gave me to try to find the equation of the hyperbola. I will gladly and eagerly read everything helpful I see in response to this. If someone else has already answered, feel free to try to answer anyway because chances are I will need several different ways of seeing it to understand it. Thank you so much to those of you who take the time to try to understand why I'm confused!

(I double checked the all the numbers that I typed above are 100% accurate to what the book is showing me)

 

[Deleted member 4993]

I have always had trouble grasping proof behind why algebraic problems work, and it's resulted in me being somewhat unable to think flexibly when trying to find certain variables. The question I need help understanding the process of solving is: Find the equation of the hyperbola given the following: foci (0, +or-8) and asymptotes y=+or-1/2x

I looked in the back of the book, and the solution is 5y^2/64 - 5x^2/256 = 1, but I can't for the life of me figure out how to get to that solution. I know that c=+or-8 and that the asymptotes are of the form y=+or-(a/b)x. My teacher told me because the asymptote gives a & b in terms of y=+or-(a/b)x, I should view 1 & 2 as a & b respectively. My teacher then said that because a/b=1/2 in this case, I needed to change it to solve for either a or b, and he wrote down a=1/2b and told me I will need to substitute that somewhere to proceed finding the equation for the hyperbola, but not only do I not know where to substitute it, I do not know how he solved it that way or why. If 1=a, and 2=b, how did he algebraically manipulate it to read a=1/2b? It seems like he added variables a and b into an equation that already had numbers to stand for variables a & b.

I hope I articulated my confusion well enough. Once again, I listed the answer to this equation according to the back of my book, as well as what the original problem gave me to try to find the equation of the hyperbola. I will gladly and eagerly read everything helpful I see in response to this. If someone else has already answered, feel free to try to answer anyway because chances are I will need several different ways of seeing it to understand it. Thank you so much to those of you who take the time to try to understand why I'm confused!

(I double checked the all the numbers that I typed above are 100% accurate to what the book is showing me)

The equation of hyperbola:

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

the equations for assymptotes (what is meant by assymptote)

\(\displaystyle y \ = \ \pm\dfrac{a}{b} \ * \ x\).....................................................................................................(2)

If the co-ordinates of the focii are (±c,0) then

c² = a² + b²..............................................................................................(3) assume that equation 2 is given to you as y = ± m * x

then

m = a/b

then

a = m * b....................................................................(4)

using (4) in (3)

c² = m² * b² + b² = b² (1+m²) → b = c/√(1+m²) .....................(5)

using (5) in (4) → a = m * c/√(1+m²) .....................(6)

Now use (5) and (6) in (1) to calculate the equation for the hyperbola.

 

[wjm11]

I have always had trouble grasping proof behind why algebraic problems work, and it's resulted in me being somewhat unable to think flexibly when trying to find certain variables. The question I need help understanding the process of solving is: Find the equation of the hyperbola given the following: foci (0, +or-8) and asymptotes y=+or-1/2x

I looked in the back of the book, and the solution is 5y^2/64 - 5x^2/256 = 1, but I can't for the life of me figure out how to get to that solution. I know that c=+or-8 and that the asymptotes are of the form y=+or-(a/b)x. My teacher told me because the asymptote gives a & b in terms of y=+or-(a/b)x, I should view 1 & 2 as a & b respectively. My teacher then said that because a/b=1/2 in this case, I needed to change it to solve for either a or b, and he wrote down a=1/2b and told me I will need to substitute that somewhere to proceed finding the equation for the hyperbola, but not only do I not know where to substitute it, I do not know how he solved it that way or why. If 1=a, and 2=b, how did he algebraically manipulate it to read a=1/2b? It seems like he added variables a and b into an equation that already had numbers to stand for variables a & b.

I hope I articulated my confusion well enough. Once again, I listed the answer to this equation according to the back of my book, as well as what the original problem gave me to try to find the equation of the hyperbola. I will gladly and eagerly read everything helpful I see in response to this. If someone else has already answered, feel free to try to answer anyway because chances are I will need several different ways of seeing it to understand it. Thank you so much to those of you who take the time to try to understand why I'm confused!

(I double checked the all the numbers that I typed above are 100% accurate to what the book is showing me)

You may also find the article and picture here: http://www.purplemath.com/modules/hyperbola.htm
to be helpful.

 

[poisonfartdrogs]

Thank you

Thanks all who replied. I have tried purplemath many a time, but for some reason it was never as helpful to me as I expected. It's probably simply because I'm so horribly bad at math, but I think I understand how to do this now. The first response was fairly clear, the step numbering helped me immensely.