Chapter 8.3, Problem 75E

Solution

The reasons why the given conclusion cannot be true.

The reasons why the given conclusion cannot be true, have been provided.

Given:

The conclusion:

A hyperbola consists of two parabolas, one on each nappe.

Concept used:

Eccentricity is defined as the distance from any point on the conic section to the focus divided by the perpendicular distance from that point to the nearest directrix.

Calculation:

The eccentricity of parabolas is always $1$ and the eccentricity of hyperbolas is always greater than $1$ .

As discussed, eccentricity is defined as the distance from any point on the conic section to the focus divided by the perpendicular distance from that point to the nearest directrix.

This implies that any point on a parabola is equidistant from the focus and the directrix, while any point on a hyperbola is nearer to the directrix than to the focus.

Moreover, hyperbolas are asymptotic to straight lines, while parabolas have tangent at any angle (except parallel to the axis); which implies that parabolas are not asymptotic to any straight lines.

Thus, the given conclusion that a hyperbola consists of two parabolas, one on each nappe; cannot be true.

Conclusion:

The reasons why the given conclusion cannot be true, have been provided.