Chapter 11, Problem 8RCC

a.

To determine

To Write: The polar equations which represent a conic with eccentricity ‘e’.

Answer to Problem 8RCC

Let d be positive number four conics of eccentricity e with focus at polar equation.

Directrix: $-x=d\Rightarrow r=\frac{ed}{1+e\cos \theta },x=-d\Rightarrow r=\frac{ed}{1-e\cos \theta }$

Directrix: $-y=d\Rightarrow r=\frac{ed}{1+e\sin \theta },y=-d\Rightarrow r=\frac{ed}{1-e\sin \theta }$

Explanation of Solution

Given information:

All the conics are in polar form.

Calculation:

Conics are formed by three things

  $\Rightarrow$ Focus

  $\Rightarrow$ Directrix

  $\Rightarrow$ Ratio of distances of each to a point on the graph

  

Here $l$ be the fixed line in the plane

F = fixed point

According to conics now:

L $=$ directrix of the conic

F $=$ focus of the conic

Consider F is located of the pole and equation of directrix ‘l’ is $x=d\left(d>0\right)$

P has Cartesian coordinates $\left(x,y\right)$

P has polar coordinates $\left(r,\theta \right)$

Now:-

A collections of points P in the plane such that $e=\frac{d\left(P,F\right)}{d\left(P,l\right)}$ is a fixed positive number is called a conic section.

Here e = eccentricity of the conic

  $d\left(P,F\right)\Rightarrow$ distance between P and F

  $d\left(P,l\right)\Rightarrow$ distance of P and l

F = Focus of the conic

L = directrix of the conic

Now:-

  $\begin{array}{l}d,\left(P,F\right)=ed\left(P,l\right)\\ d\left(P,F\right)=r\\ d\left(P,l\right)=\left(d-r\cos \theta \right)\\ r=e\left(d-r\cos \theta \right)\\ r=\frac{ed}{1+e\cos \theta }\end{array}$

If $x=-d$

Then $=\frac{ed}{1-e\cos \theta }$

If $y=d$

Then $r=\frac{ed}{1+e\sin \theta }$

b.

To determine

To Find: The type of conic based on eccentricity

Answer to Problem 8RCC

  $\Rightarrow$ If $0<e<1$ ,then the conic is ellipse

  $\Rightarrow$ If $e=1$ ,then the conic is parabola

  $\Rightarrow$ If $e>1$ ,then the conic is hyperbola

Explanation of Solution

Given information:

Considering the equations are in polar form.

Calculation:

  

A collection of points ‘P’ in the plane such that $e=\frac{d\left(P,F\right)}{d\left(P,l\right)}$ is a fixed positive number is called conic − section. Here e is eccentricity of the conic. The ‘l’ is directrix and the point = F focus of the conic.

  $d\left(P,F\right)\Rightarrow$ distance between P and F

  $d\left(P,l\right)\Rightarrow$ distance of P and l

If $0<e<1$ , , then the conic is ellipse

If $e=1$ ,then the conic is parabola

If $e>1$ ,then the conic is hyperbola