Question

What are the equations of the asymptotes of the hyperbola?
(Need ASAP)
Thank You

Answer 1

The point-slope form of the asymptote of the hyperbola is :

y - 1 = + (3/10)(x - 2)

y - 1 = - (3/10)(x - 2)

We have the equation of Hyperbola as :

⇒ [ ( y - 1 )² / 9 ] - [ ( x - 2 )² / 100 ] = 1

it can be rewritten as :

⇒ [ ( y - 1 )² / 3² ] - [ ( x - 2 )² / 10² ] = 1

and the general form of a hyperbola is given as :

⇒ [ ( y - k )² / a² ] - [ ( x - h )² / b² ] = 1

Comparing this general form with our given equation we get,

⇒ k = 1 ; a = 3 ; h = 2 ; b = 10

and the equation of the asymptote of a hyperbola is :

⇒ y - k = + (a/b)(x - h)

y - k = - (a/b)(x - h)

equating the values of k, a, h, b in the above equation we get,

⇒ y - 1 = + (3/10)(x - 2)

y - 1 = - (3/10)(x - 2)

Answer 2

`y-1=(3)/(10)(x-2)`

`y-1=-(3)/(10)(x-2)`

Explanation:

Given equation of a hyperbola:

`((y-1)^2)/(9)-((x-2)^2)/(100)=1`

As the squared term associated with y is positive, and the squared term associated with x is negative, this is a vertical hyperbola.

`\boxed{\begin{minipage}{7.4 cm}\underline{Standard equation of a vertical hyperbola}\\\\$((y-k)^2)/(a^2)-((x-h)^2)/(b^2)=1$\\\\where:\\\phantom{ww}$\bullet$ $(h,k)$ is the center. \\ \phantom{ww}$\bullet$ $(h, k\pm a)$ are the vertices. \\\phantom{ww}$\bullet$ $(h\pm b, k)$ are the co-vertices. \\\phantom{ww}$\bullet$ $(h, k\pm c)$ are the foci where $c^2=a^2+b^2$\\\phantom{ww}$\bullet$ $y -k= \pm \left((a)/(b)\right)(x-h)$ are the asymptotes.\\\end{minipage}}`

If we compare the given equation to the standard equation of a vertical hyperbola, we get:

• 
  `h = 2`
• 
  `k = 1`
• 
  `a^2 = 9 \implies a = 3`
• 
  `b^2 = 100 \implies b = 10`

The asymptotes of a hyperbola are straight lines that pass through the center (h, k) of the hyperbola. The slopes of the lines are ±a/b. Therefore, the equation for the asymptotes of a vertical hyperbola (in point-slope form) is:

`y-k=\pm (a)/(b)(x-h)`

Substitute the value of h, k, a and b:

`y-1=\pm (3)/(10)(x-2)`

Therefore, the equations of the asymptotes of the given hyperbola in point-slope form are:

`\boxed{y-1=(3)/(10)(x-2)}` and
`\boxed{y-1=-(3)/(10)(x-2)}`