Chapter 11.6, Problem 1E

To determine

To fill: The blanks.

Answer to Problem 1E

All conics can be described geometrically by using a fixed point $F$ called the focus and a fixed line $l$ called the directrix. For a fixed positive number $e$ the set of all points $P$ satisfying

$\frac{d(P,F)}{d(P,l)}=e$.

is a conic. If $e=1$, the conic is a(n) parabola; if $e<1$, the conic is a(n) ellipse; and if $e>1$, the conic is a(n) hyperbola. The number $e$ is called the eccentricity of the conic.

Explanation of Solution

Definition used:

Definition:

“The set of all points $P$ in a plane satisfying $\frac{d(P,F)}{d(P,l)}=e$ is called a conic. Where $F$ is a fixed point, called focus, a fixed line $l$ is called directrix and $e$ is the eccentricity of the conic.

Note 1:

“The conics $r=\frac{ed}{1\pm e\sin \theta }$ and $r=\frac{ed}{1\pm e\cos \theta }$ represents,

1. (i) An ellipse if $0<e<1$.
2. (ii) A parabola if $e=1$.

A hyperbola if $e>1$.”

Calculation:

From the definition, conclude that the set of all points in a plane satisfying the condition $\frac{d(P,F)}{d(P,l)}=e$ is a conic. Where the fixed point $F$ is called focus. and the fixed line $l$ is called directrix. The fixed number $e$ is called the eccentricity.

Also, by the note above, If $e=1$, the conic is a(n) parabola; if $e<1$, the conic is a(n) ellipse; and if $e>1$, the conic is a(n) hyperbola.