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Chapter 10: Q. 45 (page 668)

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

Expert verified

Equation of the hyperbola is:(x-5)2-(y-7)23=1

Graph of the hyperbola is:

Step by step solution

01

Step 1. Given information 

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

02

Step 2. Finding the equation of the hyperbola  

Comparing the given hyperbola with the standard form,

(x-h)2a2-(y-k)2b2=1,b2=c2-a2Center:(h,k)=(4,7)Focus:(±c+h,k)=(3,7)and(7,7)⇒k=7,-c+h=3andc+h=7⇒h=7+32andc=7-32⇒h=5andc=2Vertex:(a+h,k)=(6,7)⇒a=1b2=c2-a2⇒b2=22-12⇒b2=4-1⇒b2=3⇒b=±3

Substituting h,k,aandbin the standard equation,

(x-5)21-(y-7)23=1⇒(x-5)2-(y-7)23=1

03

Step 3. Graph of the hyperbola 

Equations of asymptotes:

(y-k)=±ba(x-h)⇒(y-7)=±3(x-5)

Graph of the hyperbola is:

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