Definition via focus, directrix, eccentricity e: Question bank

Recall the one rule everything tests: a point $P$ is on the conic exactly when $\frac{P F}{P M} = e,$ where $P F$ is the straight-line distance from $P$ to the focus $F$ , and $P M$ is the perpendicular distance from $P$ to the directrix line $L$ . Keep that picture — a dot $F$ , a wall $L$ — in front of you for every question.

True or false — justify

The eccentricity

e

is measured in the same units as

P F

and

P M

(metres, say).

False —

e = P F / P M

is a ratio of two lengths, so the units cancel and

e

is a pure number; that is exactly why it describes shape and not size.

Every conic has both a focus and a directrix.

True in the focus–directrix sense, but the circle (

e = 0

) is the degenerate limit where the directrix is pushed infinitely far away, so we usually describe a circle by its centre alone instead.

e = 0.5

, then a point on the curve is always exactly half as far from the focus as from the directrix.

True —

e = 0.5

literally means

P F = \frac{1}{2} P M

at every single point on the curve, which is what forces the closed oval (ellipse) shape.

Two ellipses with the same

e

must be the same size.

False —

e

fixes the "squashedness" only; a coin-sized and a stadium-sized ellipse can both have

e = 0.6

because you can scale the whole figure and the ratio

P F / P M

is unchanged.

The focus is allowed to lie on the directrix.

False — the definition requires

F \in / L

; if

F

were on

L

, points on the line would have

P F = P M = 0

and the ratio would be undefined, collapsing the whole construction.

A parabola (

e = 1

) is "between" an ellipse and a hyperbola.

True in the sense that

e = 1

is the exact boundary: for

e

just below

1

the curve barely closes (ellipse) and for

e

just above

1

it barely opens into two branches (hyperbola), with the parabola as the knife-edge case.

Increasing

e

from

0.9

2

moves the curve smoothly and continuously through the parabola.

True — the single dial

e

deforms the curve continuously; as

e

passes through

1

the ellipse's far end "opens up to infinity" and becomes the second branch of a hyperbola.

For a hyperbola, only the branch nearest the focus satisfies

P F / P M = e

False — both branches satisfy the same ratio with the same focus and directrix; the far branch simply has

P F

larger than

P M

(which

e > 1

permits), so it still obeys the rule.

Spot the error

"To get

P M

for the directrix

x = - d

, I use

(x + d)^{2} + y^{2}

." — find the mistake.

That formula is the distance to the point

(- d, 0)

, not to the line. Perpendicular distance to the vertical line

x = - d

∣ x + d ∣

, ignoring

y

entirely.

"From

x^{2} + y^{2} = e ∣ x + d ∣

, I square to get

x^{2} + y^{2} = e^{2} (x + d)^{2}

, so squaring lost nothing." — is this safe here?

It is safe in this case because both sides are non-negative distances, but the general warning stands: squaring can create phantom solutions and it silently absorbs the absolute value, so you must remember both sides were

\geq 0

"Since

e

is 'eccentricity', a large

e

means a large, spread-out ellipse." — spot the confusion.

An ellipse only exists for

0 < e < 1

, so there is no "large

e

" ellipse; and even within that range

e

controls how flattened the oval is, not how big it is.

"For

e = 1

the equation

x^{2} + y^{2} = e^{2} (x + d)^{2}

still has an

x^{2}

term, so it's a genuine second-degree curve in

x

." — where's the slip?

Setting

e = 1

gives

x^{2} + y^{2} = x^{2} + 2 d x + d^{2}

; the

x^{2}

cancels on both sides, leaving

y^{2} = 2 d x + d^{2}

, which is linear in

x

— that vanishing

x^{2}

is precisely what makes it a parabola.

"The directrix distance from

P = (x, y)

to the line

x = 4

∣ x + 4∣

." — correct it.

The gap to the line

x = 4

∣ x - 4∣

; you subtract the line's

x

-value. Only the line

x = - 4

gives

∣ x + 4∣

"An ellipse and its directrix don't touch, but a big enough ellipse could reach across and cross the directrix." — true?

False — for

0 < e < 1

every point satisfies

P F = e P M

with

P M > 0

, so

P M

is never zero; the curve can never meet the directrix, no matter how large.

"Because

e = c / a

, I can find

e

for a parabola using its

c

and

a

." — what's wrong?

A parabola has no centre, so there is no

c

(centre-to-focus) or

a

(centre-to-vertex) in that sense; the

e = c / a

formula only applies to ellipses and hyperbolas, and a parabola simply has

e = 1

by definition.

Why questions

Why does

0 < e < 1

force the curve to close up into an ellipse?

Because

P F = e P M

with

e < 1

keeps the point permanently closer to the focus than to the directrix; a point can never wander infinitely far without

P F

eventually exceeding

e P M

, so the locus is bounded and wraps around.

Why does

e > 1

produce two separate branches instead of one closed curve?

With

e > 1

the point may be farther from the focus than from the directrix, so as

x

grows one squared term is subtracted from the other; the equation admits arbitrarily large

∣ y ∣

, and the curve escapes to infinity on both sides of the directrix — two open branches.

Why is

P M

a perpendicular distance and not just any distance to the line?

"Distance from a point to a line" means the shortest possible distance, and the shortest path from a point to a line is always the perpendicular; any slanted measurement would be longer and would make the ratio

P F / P M

depend on an arbitrary choice.

Why does the coefficient

(1 - e^{2})

x^{2}

tell you the conic type at a glance?

Its sign decides the geometry:

(1 - e^{2}) > 0

(i.e.

e < 1

) makes both squared terms positive → closed ellipse;

(1 - e^{2}) = 0

(

e = 1

) kills the term → parabola;

(1 - e^{2}) < 0

(

e > 1

) flips the sign → opposite-signed terms → hyperbola.

Why do we usually place the focus at the origin when deriving the equation?

It makes

P F = x^{2} + y^{2}

the cleanest possible expression (no shifts to expand), so the algebra stays readable and the role of

e

and

d

stays visible; any other placement gives the same shape, just with extra constants.

Why is the circle called a limiting case rather than a normal member of the family?

e \to 0

the ratio

P F / P M \to 0

forces

P F \to 0

, so the focus collapses onto the centre and the directrix must recede to infinity to keep

P M

finite; the directrix disappears from the picture, so the focus–directrix machinery degenerates.

Why must both sides of

x^{2} + y^{2} = e ∣ x + d ∣

be non-negative before squaring?

A square root is defined as the non-negative value and

∣ x + d ∣ \geq 0

, and

e > 0

, so both sides are genuinely

\geq 0

; this is the condition that lets squaring preserve the equation without introducing sign-flipped false solutions.

Edge cases

What conic do you get exactly at

e = 1

, and what makes it fragile?

A parabola: it is the unique boundary case where

P F = P M

everywhere. A tiny nudge of

e

either way turns it into an ellipse or a hyperbola, so it's the "knife-edge" of the family.

What happens to the locus as

e \to 0^{+}

The point is squeezed ever closer to the focus (

P F = e P M \to 0

), so the ellipse shrinks toward a single point unless the directrix is simultaneously sent to infinity — the limit is the circle centred at the focus.

What happens to the ellipse's shape as

e \to 1^{-}

It becomes more and more elongated (a "cigar"); the far vertex races away and in the limit the closed oval opens up, handing off to the parabola.

Can

e

be negative or exactly equal to some value that gives no curve at all?

No —

e

is defined as a ratio of two distances, both non-negative, so

e \geq 0

always; there is no value of

e \geq 0

that yields an empty locus, since you can always find points satisfying the ratio.

If the directrix passed through the focus, what would the "conic" look like?

The construction breaks: points on the shared line would give

P F = P M = 0

and an undefined ratio, so this configuration is explicitly forbidden (

F \in / L

) rather than producing a special curve.

For a point sitting exactly on the directrix, what is

P M

, and can such a point ever be on a real conic?

P M = 0

there, forcing

P F = e \cdot 0 = 0

, i.e. the point would also have to be at the focus — impossible since

F \in / L

; so no genuine conic point ever lies on its own directrix.

If you double both

P F

and

P M

for every point (a uniform scaling), does

e

change?

No — the ratio

P F / P M

is unchanged under uniform scaling, which is exactly why

e

is a shape invariant and a scaled-up conic keeps the same eccentricity.

Connections

Parabola - Standard equation y^2=4ax
Ellipse - Standard form and c^2 = a^2 - b^2
Hyperbola - Standard form and c^2 = a^2 + b^2
Circle as limiting case e=0
Distance formula and distance from a point to a line
Latus rectum and semi-latus rectum