Answer
a) See the explanation
b) See the explanation.
c) See the explanation.
Work Step by Step
a)
Let us consider the points $A$ on the conic section, whose focus is $f$ and directrcix is $L$. Thus, the eccentricity$(e)$ can be calculated as:
$e=\dfrac{|Af|}{|AL|}$; where, $\dfrac{|Af|}{|AL|}$ is a fixed ratio.
b)
For an ellipse: $e \lt 1$
For a hyperbola: $e \gt 1$
For a parabola: $e = 1$
c)
For a conic section with eccentricity $(e)$ and directrx $x=d$, the polar equation can be represented as:
$r=\dfrac{ed}{(1+e \cos \theta)}$; for directrix: $x=d$;
$r=\dfrac{ed}{(1-e \cos \theta)}$; for directrix: $x=-d$;
$r=\dfrac{ed}{(1+e \sin \theta)}$; for directrix: $y=d$;
$r=\dfrac{ed}{(1-e \sin \theta)}$; for directrix: $y=-d$;
