Chapter 8.3, Problem 63E

Solution

If the given statement is true or false.

It has been determined that the given statement is true.

Given:

The distance from a focus of a hyperbola to the closer vertex is $a\left(e-1\right)$ , where $a$ is the semi-transverse axis and $e$ is the eccentricity.

Concept used:

The foci of a horizontal hyperbola with center at origin are $\left(\pm c,0\right)$ and vertices are $\left(\pm a,0\right)$ , where $c>a>0$ .

The eccentricity of a hyperbola is given as $e=\frac{c}{a}$ .

Calculation:

It is given that $e=\frac{c}{a}$ .

Put $e=\frac{c}{a}$ in $a\left(e-1\right)$ to get,

  $a\left(e-1\right)=a\left(\frac{c}{a}-1\right)=c-a$

Now, the foci of a horizontal hyperbola with center at origin are $\left(\pm c,0\right)$ and vertices are $\left(\pm a,0\right)$ , where $c>a>0$ .

Then, the nearest vertex to the focus $\left(c,0\right)$ is $\left(a,0\right)$ and to the focus $\left(-c,0\right)$ is $\left(-a,0\right)$ .

In either case, the distance between a focus and its nearest vertex is $c-a$ .

This shows that the distance from a focus of a horizontal hyperbola with center at origin, to the closer vertex is $a\left(e-1\right)$ , where $a$ is the semi-transverse axis and $e$ is the eccentricity.

Now, distance is preserved if the orientation of a figure is changed or in case of translation.

This implies that the given statement is true for any hyperbola.

Conclusion:

It has been determined that the given statement is true.