a.
To show: that the following equation represents a hyperbola using rotation of axis.
a.
Answer to Problem 33E
Explanation of Solution
Given information: Given equation is
Calculation:
Need to use rotation of axes to show that the equation,
For that first have to eliminate xy term from the given equation.
On comparing the given equation with:
Rotate the axes through the acute angle
The Rotation of Axes Formula then give:
Substituting into the given equation:
Therefore,
Hence Proved.
b.
To find: the XY − and xy - coordinates of the center, vertices and foci.
b.
Answer to Problem 33E
Center is (5, 0).
Vertices are (6, 0), (4, 0).
Foci are
Explanation of Solution
Given information: Given hyperbola is
Calculation:
The vertices ( h+a ,k ) ,( h-a ,k ) are the two bending points of the hyperbola with center (h ,k) and semi axis a,b.
Given parabola is right-left hyperbola with center ( h , k ) as (5,0) and semi axis a =1 and semi −conjugate −axis b =1.
So, the vertices are (5+1,0) ,(5-1,0) = (6,0) ,(4,0).
Therefore vertices are (6, 0) , (4,0).
For a right −left facing hyperbola, the foci are defined as ( h+c ,k ) ,( h-c ,k ), where
So Foci are
c.
To find: the equation of the asymptotes in XY −and xy −coordinates.
c.
Answer to Problem 33E
Equations of asymptotes are:
Explanation of Solution
Given information: Given hyperbola is
Formula used:
For right-left hyperbola the asymptotes are
Calculation:
The asymptotes are the lines the hyperbola tends to at
For right-left hyperbola the asymptotes are
So, equations of asymptotes are:
Chapter 11 Solutions
Precalculus: Mathematics for Calculus - 6th Edition
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