Chapter 9.3, Problem 5E

(a)

To determine

To check:The hyperbola equation $\frac{{\left(x-h\right)}^2}{a^2}-\frac{{\left(y-k\right)}^2}{b^2}=1$ has horizontal transverse axis or vertical transverse axis.

Answer to Problem 5E

The hyperbola equation $\frac{{\left(x-h\right)}^2}{a^2}-\frac{{\left(y-k\right)}^2}{b^2}=1$ has horizontal transverse axis.

Explanation of Solution

Given information:

The hyperbola equation is $\frac{{\left(x-h\right)}^2}{a^2}-\frac{{\left(y-k\right)}^2}{b^2}=1$ .

Calculation:

If $x$ goes first in the equation, the equation has a horizontal transverse axis, and if $y$ goes first in the equation, the equation has a vertical transverse axis.

In the hyperbola equation $\frac{{\left(x-h\right)}^2}{a^2}-\frac{{\left(y-k\right)}^2}{b^2}=1$ , $x$ comes first. So, this hyperbola equation has horizontal transverse axis.

Therefore, the hyperbola equation $\frac{{\left(x-h\right)}^2}{a^2}-\frac{{\left(y-k\right)}^2}{b^2}=1$ has horizontal transverse axis.

(b)

To determine

To check: The hyperbola equation $\frac{{\left(y-k\right)}^2}{a^2}-\frac{{\left(x-h\right)}^2}{b^2}=1$ has horizontal transverse axis or vertical transverse axis.

Answer to Problem 5E

The hyperbola equation $\frac{{\left(y-k\right)}^2}{a^2}-\frac{{\left(x-h\right)}^2}{b^2}=1$ has vertical transverse axis.

Explanation of Solution

Given information:

The hyperbola equation is $\frac{{\left(y-k\right)}^2}{a^2}-\frac{{\left(x-h\right)}^2}{b^2}=1$ .

Calculation:

If $x$ goes first in the equation, the equation has a horizontal transverse axis, and if $y$ goes first in the equation, the equation has a vertical transverse axis.

In the hyperbola equation $\frac{{\left(y-k\right)}^2}{a^2}-\frac{{\left(x-h\right)}^2}{b^2}=1$ , $y$ comes first, so this hyperbola equation has vertical transverse axis.

Therefore, the hyperbola equation $\frac{{\left(y-k\right)}^2}{a^2}-\frac{{\left(x-h\right)}^2}{b^2}=1$ has vertical transverse axis.

(c)

To determine

To check:The hyperbola equation $\frac{y^2}{a^2}-\frac{x^2}{b^2}=1$ has horizontal transverse axis or vertical transverse axis.

Answer to Problem 5E

The hyperbola equation $\frac{y^2}{a^2}-\frac{x^2}{b^2}=1$ has vertical transverse axis.

Explanation of Solution

Given information:

The hyperbola equation is $\frac{y^2}{a^2}-\frac{x^2}{b^2}=1$ .

Calculation:

If $x$ goes first in the equation, the equation has a horizontal transverse axis, and if $y$ goes first in the equation, the equation has a vertical transverse axis.

In the hyperbola equation $\frac{y^2}{a^2}-\frac{x^2}{b^2}=1$ , $y$ comes first, so this hyperbola equation has vertical transverse axis.

Therefore, the hyperbola equation $\frac{y^2}{a^2}-\frac{x^2}{b^2}=1$ has vertical transverse axis.

(d)

To determine

To check:The hyperbola equation $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ has horizontal transverse axis or vertical transverse axis.

Answer to Problem 5E

The hyperbola equation $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ has horizontal transverse axis.

Explanation of Solution

Given information:

The hyperbola equation is $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ .

Calculation:

If $x$ goes first in the equation, the equation has a horizontal transverse axis, and if $y$ goes first in the equation, the equation has a vertical transverse axis.

In the hyperbola equation $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ , $x$ comes first, so this hyperbola equation has horizontal transverse axis.

Therefore, the hyperbola equation $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ has horizontal transverse axis.