Chapter 9.3, Problem 4E

a.

To determine

To find: Which equation represents a hyperbola with a horizontal axis and vertical axis.

Answer to Problem 4E

The given equation represents hyperbola with horizontal transverse axis.

Explanation of Solution

Given information:

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

Calculation:

The standard form of equation of hyperbola with centre at $\left(h,k\right)$ and horizontal transverse axis is

  $\frac{{\left(x-h\right)}^2}{a^2}-\frac{{\left(y-k\right)}^2}{b^2}=1$ and

The standard form of equation of hyperbola with centre at $\left(h,k\right)$ and vertical transverse axis is

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

The given equation is

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

Therefore, the given equation represents hyperbola with horizontal transverse axis.

Hence,

The given equation represents hyperbola with horizontal transverse axis.

b.

To determine

To find: Which equation represents a hyperbola with a horizontal axis and vertical axis.

Answer to Problem 4E

The given equation represents hyperbola with vertical transverse axis.

Explanation of Solution

Given information:

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

Calculation:

The standard form of equation of hyperbola with centre at $\left(h,k\right)$ and horizontal transverse axis is

  $\frac{{\left(x-h\right)}^2}{a^2}-\frac{{\left(y-k\right)}^2}{b^2}=1$ and

The standard form of equation of hyperbola with centre at $\left(h,k\right)$ and vertical transverse axis is

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

The given equation is

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

Therefore, the given equation represents hyperbola with vertical transverse axis.

Hence,

The given equation represents hyperbola with vertical transverse axis.

c.

To determine

To find: Which equation represents a hyperbola with a horizontal axis and vertical axis.

Answer to Problem 4E

The given equation represents hyperbola with vertical transverse axis.

Explanation of Solution

Given information:

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

Calculation:

The standard form of equation of hyperbola with centre at $\left(h,k\right)$ and horizontal transverse axis is

  $\frac{{\left(x-h\right)}^2}{a^2}-\frac{{\left(y-k\right)}^2}{b^2}=1$ and

The standard form of equation of hyperbola with centre at $\left(h,k\right)$ and vertical transverse axis is

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

The given equation is

  $\frac{y^2}{a^2}-\frac{x^2}{b^2}=1$

Therefore, the given equation represents hyperbola with vertical transverse axis.

Hence,

The given equation represents hyperbola with vertical transverse axis.

d.

To determine

To find: Which equation represents a hyperbola with a horizontal axis and vertical axis.

Answer to Problem 4E

The given equation represents hyperbola with horizontal transverse axis.

Explanation of Solution

Given information:

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

Calculation:

The standard form of equation of hyperbola with centre at $\left(h,k\right)$ and horizontal transverse axis is

  $\frac{{\left(x-h\right)}^2}{a^2}-\frac{{\left(y-k\right)}^2}{b^2}=1$ and

The standard form of equation of hyperbola with centre at $\left(h,k\right)$ and vertical transverse axis is

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

The given equation is

  $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

Therefore, the given equation represents hyperbola with horizontal transverse axis.

Hence,

The given equation represents hyperbola with horizontal transverse axis.