Chapter SH, Problem 9.13EP

To determine

To find: the equation of the hyperbola shown in the graph.

Answer to Problem 9.13EP

The equation of the hyper bola her is ${\text{16y}}^{\text{2}}-{\text{9x}}^{\text{2}}\text{=144}$ .

Explanation of Solution

Given:

  

Concept used:

The equation of hyperbola having foci at vertical axis is:

  $\frac{{\text{y}}^{\text{2}}}{{\text{a}}^{\text{2}}}\text{-}\frac{{\text{x}}^{\text{2}}}{{\text{b}}^{\text{2}}}\text{=1}$ .

  ${\text{c}}^{\text{2}}{\text{=a}}^{\text{2}}{\text{+b}}^{\text{2}}$ .

Vertices is denoted by $\text{a}$ .

Co-vertices is denoted by $\text{b}$ .

Foci is denoted by $\text{c}$ .

Calculation:

The equation of hyperbola having foci at vertical axis is:

  $\frac{{\text{y}}^{\text{2}}}{{\text{a}}^{\text{2}}}\text{-}\frac{{\text{x}}^{\text{2}}}{{\text{b}}^{\text{2}}}\text{=1}$ …………….. $\left(\text{i}\right)$

Asymptotes will cross at $\left(0,0\right)$ .

Foci $\text{=}\left(\text{0,c}\right)\text{,}\left(\text{0,-c}\right)=\left(0,5\right),\left(0,-5\right)$ .

Vertices $=\left(\text{0,a}\right)\text{,}\left(\text{0,-a}\right)=\left(0,3\right),\left(0,-3\right)$ .

Co- vertices $\text{=}\left(\text{0,b}\right)\text{,}\left(\text{0,-b}\right)$ .

Co-vertices can be calculated as:

  ${\text{c}}^{\text{2}}{\text{=a}}^{\text{2}}{\text{+b}}^{\text{2}}$ .

  $\Rightarrow {\text{5}}^{\text{2}}{\text{=3}}^{\text{2}}{\text{+b}}^{\text{2}}$ .

  $\Rightarrow {\text{25-9=b}}^{\text{2}}$ .

  ${\text{b}}^{\text{2}}=16$ …… $\left(\text{ii}\right)$

Similarly, ${\text{a}}^2=9$ …. $\left(\text{iii}\right)$

Putting equation $\left(\text{ii}\right)$ and $\left(\text{iii}\right)$ in equation $\left(\text{i}\right)$ :

  $\frac{{\text{y}}^{\text{2}}}{{\text{a}}^{\text{2}}}\text{-}\frac{{\text{x}}^{\text{2}}}{{\text{b}}^{\text{2}}}\text{=1}$ .

  $\frac{{\text{y}}^{\text{2}}}{\text{9}}\text{-}\frac{{\text{x}}^{\text{2}}}{\text{16}}\text{=1}$ .

  ${\text{16y}}^{\text{2}}-{\text{9x}}^{\text{2}}\text{=144}$ .

Hence, the equation of the hyper bola her is ${\text{16y}}^{\text{2}}-{\text{9x}}^{\text{2}}\text{=144}$ .