Definition via focus, directrix, eccentricity e: Foundations

This page assumes you have seen nothing. We name every symbol, draw the picture behind it, and say why the topic can't move without it. Read top to bottom; each brick sits on the one before.

0. Points and coordinates — the dot with an address

Figure — Definition via focus, directrix, eccentricity e

Look at the figure. The horizontal ruler is the == $x$ -axis, the vertical ruler is the $y$ -axis, and where they cross is the origin== $(0, 0)$ . Every dot on the page has exactly one address $(x, y)$ .

Why the topic needs this. The parent note writes $P = (x, y)$ , $F = (0, 0)$ , $F = (1, 0)$ . Those are all just dots-with-addresses. If you can't read an address, you can't place the focus or the moving point.

1. The line, and one very special kind: the vertical line $x = - d$

In the figure the amber line is $x = - d$ (here $d > 0$ , so the line sits to the left of the origin). Notice every dot on it shares the same $x$ -value; only $y$ changes as you slide up and down. That is exactly why the line is vertical.

Why the topic needs this. The parent's directrix is the line $x = - d$ (and later $x = 4$ , $x = 1$ ). The directrix is always one of these vertical lines in the derivation, so you must be comfortable that " $x = - d$ " is a whole line, not a single point.

2. The square root symbol and squares $x^{2}$

Why the topic needs this. Distances are built from squares (next section), and a square root is how we undo those squares to recover a plain length. Also: $x^{2}$ is never negative, because a number times itself is positive whether the number was positive or negative. That single fact — "a square can't be negative" — is what later forces an ellipse to close up and a hyperbola to fly apart.

Recall Why is

never negative? By agreement $n$ means the non-negative answer ::: so a length like $P F = x^{2} + y^{2}$ is always $\geq 0$ , which is exactly what we want for a distance.

3. Distance between two points — the ruler formula

What it looks like. Drop from $P$ across to $Q$ and you make a right triangle: the horizontal leg is the gap in $x$ (length $∣ x - a ∣$ ), the vertical leg is the gap in $y$ (length $∣ y - b ∣$ ), and the slanted side joining $P$ to $Q$ is the distance we want. Pythagoras says leg $^{2}$ + leg $^{2}$ = hypotenuse $^{2}$ , so the hypotenuse is the square root of the two squared gaps. That IS the formula.

Why the topic needs this. The parent writes $P F = x^{2} + y^{2}$ . That is just this formula with $Q = F = (0, 0)$ : the gaps are $(x - 0)$ and $(y - 0)$ . Every "distance to the focus" line in every worked example is one use of this one triangle. See Distance formula and distance from a point to a line for the full toolkit.

4. Perpendicular distance to a line — a DIFFERENT ruler

Distance to a point and distance to a line are not the same measurement, and confusing them is the parent note's headline mistake.

What it looks like. In the figure, from $P$ we could reach the line by many slanted paths (grey), but the cyan path — the one that hits the line square-on — is the shortest. For the vertical line $x = - d$ , the perpendicular is horizontal, so the only thing that matters is the horizontal gap between $x$ and $- d$ : $P M = ∣ x - (- d) ∣ = ∣ x + d ∣$ The $y$ -coordinate of $P$ plays no role at all — slide $P$ straight up or down and its perpendicular distance to a vertical line doesn't change.

Why the topic needs this. This is $P M$ , the denominator of the whole conic definition. If you wrongly used $x^{2} + y^{2}$ here, you would be measuring to a point, not the line, and every conic would collapse.

5. Absolute value $∣ \cdot ∣$ — distance is never negative

Why the topic needs this. In $P M = ∣ x + d ∣$ the bars guarantee the gap is a genuine (non-negative) distance whether $P$ sits left or right of the directrix. Later, squaring both sides quietly erases these bars — because $∣ x + d ∣^{2} = (x + d)^{2}$ — which is exactly why the parent can drop the bars after squaring.

Recall Why does squaring remove the

$∣ \cdot ∣$ ? Because $∣ a ∣^{2} = a^{2}$ for every $a$ ::: squaring both a positive and a negative value gives the same non-negative square, so the bars become unnecessary once you square.

6. The ratio and the letter $e$ — the personality dial

What it looks like. Stand at $P$ . Measure your walk to the focus, then your perpendicular walk to the directrix, and divide. Keep only the dots where that division gives the same answer $e$ every time — those dots trace the conic.

Why the topic needs this. This single number is the whole subject. $e < 1$ (must stay closer to the focus) closes the curve into an ellipse; $e = 1$ (equal distances) gives a parabola; $e > 1$ gives a hyperbola; and the limit $e \to 0$ is the circle. Everything else in the parent note is bookkeeping around this ratio.

7. Reading Greek and named points ( $F$ , $L$ , $M$ , $a$ , $c$ , $d$ )

Why the topic needs this. These are just names so we can talk about the picture without pointing. Whenever the parent says " $P F$ " it means "the distance from $P$ to $F$ " — a length, built by the formula of §3. See Latus rectum and semi-latus rectum for one more named length you'll meet later.

Prerequisite map

Read it as: dots and lines feed the two distances; the two distances feed the ratio; the ratio is the conic.

Equipment checklist

Cover the right side and test yourself. If any answer is fuzzy, reread that section before the parent note.

What does

(x, y)

mean?

An address:

x

steps right,

y

steps up, from the origin

(0, 0)

What kind of object is

x = - d

A whole vertical line (every

y

, one fixed

x

), sitting

d

left of the origin when

d > 0

Give the distance from

P = (x, y)

F = (0, 0)

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

(distance formula with

F

at the origin).

Why can a square like

x^{2}

never be negative?

A number times itself is positive whether the number was positive or negative.

What is the perpendicular distance from

(x, y)

x = - d

∣ x + d ∣

— only the horizontal gap;

y

is irrelevant.

Why the absolute-value bars in

∣ x + d ∣

A distance must be

\geq 0

, so we strip the sign; squaring later removes the bars since

∣ a ∣^{2} = a^{2}

What is

e

and is it a size or a shape?

e = P F / P M

, the fixed ratio — a shape number, not a size.

Which points build a conic?

Every

P

whose ratio

P F / P M

equals the same value

e

Connections

3.4.01 Definition via focus, directrix, eccentricity e (Hinglish)
Distance formula and distance from a point to a line
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
Latus rectum and semi-latus rectum