Question

In Problems 41– 48, find an equation for the hyperbola described. Graph the equation by hand. Foci at $(3,7)$and $(7,7)$; vertex at$(6,7)$.

Short answer

Equation of the hyperbola is:${(x-5)}^2-\frac{{(y-7)}^2}{3}=1$

Graph of the hyperbola is:

Step-by-step solution

Step 1. Given information 

Hyperbola with Foci at $(3,7)$and $(7,7)$; vertex at$(6,7)$.

Step 2. Finding the equation of the hyperbola  

Comparing the given hyperbola with the standard form,

$$\begin{gathered} \frac{{(x-h)}^2}{a^2}-\frac{{(y-k)}^2}{b^2}=1,b^2=c^2-a^2 \\ \mathrm{Center}:(h,k)=(4,7) \\ \mathrm{Focus}:(\pm c+h,k)=(3,7)\mathrm{and}(7,7) \\ \Rightarrow k=7,-c+h=3\mathrm{and}c+h=7 \\ \Rightarrow h=\frac{7+3}{2}\mathrm{and}c=\frac{7-3}{2} \\ \Rightarrow h=5\mathrm{and}c=2 \\ \mathrm{Vertex}:(a+h,k)=(6,7) \\ \Rightarrow a=1 \\ {b}^2=c^2-a^2 \\ \Rightarrow b^2=2^2-1^2 \\ \Rightarrow b^2=4-1 \\ \Rightarrow b^2=3 \\ \Rightarrow b=\pm \sqrt{3} \end{gathered}$$

Substituting $h,k,a\mathrm{and}b$in the standard equation,

$$\begin{gathered} \frac{{(x-5)}^2}{1}-\frac{{(y-7)}^2}{3}=1 \\ \Rightarrow {(x-5)}^2-\frac{{(y-7)}^2}{3}=1 \end{gathered}$$

Step 3. Graph of the hyperbola 

Equations of asymptotes:

$$\begin{gathered} (y-k)=\pm \frac{b}{a}(x-h) \\ \Rightarrow (y-7)=\pm \sqrt{3}(x-5) \end{gathered}$$

Graph of the hyperbola is: