Intuition The one core ideaA hyperbola is what you trace when you insist that the difference between your distances to two fixed pins stays constant. Everything else on the parent page — a handful of letters for lengths, one number for shape, and two slanted guide-lines — is just bookkeeping that measures how that difference-curve is stretched and how steeply it flies off to infinity.
This page assumes you have seen nothing . Before you can read the parent note , every squiggle it uses must first become a picture in your head. So we build them one at a time, each new symbol resting on the previous one — no letter used before this page has defined it.
Everything happens on a flat sheet — the plane . We pin down any spot on it with two numbers.
( x , y )
x right (positive) or left (negative) of a chosen centre.y up (positive) or down (negative).The centre where both are zero is the origin , written ( 0 , 0 )
Picture: two number-lines crossing at a right angle. The horizontal one is the ==x , the vertical one is the y x y
Intuition Why the topic needs thisA hyperbola is a set of points — a rule that says "yes" or "no" to each spot on the plane. To even ask "is this point on the curve?" we must be able to name the spot, and ( x , y )
Before any distance, we name the two characters in the story.
Why the topic needs this: the whole definition of a hyperbola is a rule about the moving point P fixed pins. We must have separate names for "the thing that moves" and "the thing that stays."
To talk about "how far is P distance .
Intuition Where the distance formula comes fromPut the variable point P F
Read the pieces:
x − f 1 y − f 2 Squaring each removes any minus sign (a length can't be negative) and the square root turns the sum of squared legs back into a genuine length. That squaring-then-rooting is exactly the Pythagorean idea.
Common mistake Forgetting to square thegaps , not the coordinates
Why it slips: you see x f 1 The fix: subtract first to get the gap, then square. ( x − f 1 ) 2 x 2 − f 1 2
Why the topic needs it: the whole definition of a hyperbola is "distance to F 1 minus distance to F 2 P F
Definition Focus (plural: foci)A focus is a fixed point (the kind of F around . A hyperbola has two , written F 1 F 2 1 2
Picture: two thumbtacks stuck in the plane. In the parent's standard setup they sit on the x ( − c , 0 ) ( + c , 0 )
Why the topic needs it: no foci, no hyperbola. They are the two things whose distances we compare.
The defining rule is P F 1 − P F 2 = 2 a
Definition Absolute value∣ z ∣ ∣ z ∣ z ∣5∣ = 5 ∣ − 5∣ = 5 Picture: distance of z 0 ≥ 0
Intuition Why bars appear in the hyperbola definitionOn one branch you are closer to F 2 P F 1 − P F 2 positive . On the mirror branch you are closer to F 1 negative . The bars say "we don't care which pin is nearer — only that the gap size equals 2 a two branches instead of one.
Before we can meet a b c
Definition Transverse axis and conjugate axis
The transverse axis is the straight line through both foci. In the standard setup the foci sit on the x the transverse axis lies along the x . This is the axis the curve actually crosses — at the two vertices .
The conjugate axis is the line through the centre perpendicular to the transverse axis — so in the standard setup it lies along the y
Picture: the transverse axis is the skewer running through both thumbtacks and piercing the curve; the conjugate axis is the crossbar at 9 0 ∘
Intuition Why perpendicular, and why a "conjugate" axis at allBy symmetry the whole figure is a mirror image across the line through the foci, and across the line at right angles to it through the centre. Those two mirror-lines are exactly the transverse and conjugate axes. We keep the conjugate one because, although the curve misses it, its half-length b
These three letters are the parent's whole vocabulary. Each is a length measured along the axes we just named, and each has a picture.
Definition The three core lengths
a transverse axis — the distance from the centre out along the x vertex (where the curve crosses). The full constant difference in the defining rule is 2 a c focus (also along the transverse axis). Because the foci sit outside the vertices, c > a b conjugate axis — a length measured along the y defined by b 2 = c 2 − a 2
Intuition Why three letters and not twoa c equation looks ugly with c 2 − a 2 b b
c 2 = a 2 − b 2 Why it feels right: ellipses use minus.
The fix: hyperbola foci are farther out than vertices, so c biggest of the three: c 2 = a 2 + b 2
Why the topic needs them: distance uses x 2 y 2
a 2 x 2 bottom top a 2 x 2 how big is your horizontal reach compared with the vertex reach a At a vertex x = a 1
The standard form a 2 x 2 − b 2 y 2 = 1 your scaled horizontal reach minus your scaled vertical reach must equal one. The − 1 +
x b 2 y 2 = a 2 x 2 − 1 never be negative . That forces a 2 x 2 ≥ 1 ∣ x ∣ ≥ a only when you are at least a − a < x < a
When x ≥ a right branch (in the right half-plane).
When x ≤ − a left branch (in the left half-plane).
At exactly x = ± a y = 0
Picture: two separate arcs, one on each side of a blank vertical strip of width 2 a y
Slope = rise over run = (how far up) ÷ (how far across). A slope of a b a b Picture: the steepness of a ramp.
Definition The central rectangle (the guide-box)Mark the four corners ( ± a , ± b ) a b central rectangle , also called the guide-box . Its two diagonals are the asymptotes — the straight lines the branches chase forever. Their slope is corner-height over corner-width = a b
Why the topic needs it: the asymptotes y = ± a b x a , b
e = a c plain number with no units — a shape score.
Because c > a e > 1 e = 0 0 < e < 1 e = 1 e which conic you have and how open it is.
Why the topic needs it: e the unified conic definition that ties hyperbola, ellipse, and parabola together.
Definition Limit at infinity"As ∣ x ∣ → ∞ watch what happens when x (far out along the axis). We ask what a quantity approaches , not what it equals at any finite spot.
Intuition Why the topic needs itAsymptotes are a far-away fact: near the centre the curve is clearly bent, but very far out the "1 x 2 / a 2 Asymptotes and Limits at Infinity .
Variable point P and fixed point F
Difference rule equals 2a
Transverse and conjugate axes
Standard equation and domain
Fractions and ratio b over a
Slope and central rectangle
Related foundations reuse the same toolkit: Ellipse — standard forms, foci, eccentricity uses distance-sum , Parabola — standard forms, focus, directrix uses one focus, Rectangular Hyperbola xy=c^2 is the a = b Conic Sections from a Double Cone .
Test yourself — cover the right side.
Meaning of ( x , y ) walk x y
Difference between P F P F
Distance from P = ( x , y ) F = ( f 1 , f 2 ) What a focus is a fixed pin the curve is defined around; a hyperbola has two
Why the bars ∣ ⋅ ∣ to allow either branch — we fix the size of the difference, not its sign
What the transverse axis is the line through both foci (the x
What the conjugate axis is the line through the centre perpendicular to the transverse axis (the y
What 2 a the constant difference of the two focal distances; a
What c a centre-to-focus distance; c > a
Definition of b helper length with b 2 = c 2 − a 2
Focus relation (and how it differs from ellipse) hyperbola c 2 = a 2 + b 2 c 2 = a 2 − b 2
x 2 ∣ x ∣
Domain of x 2 / a 2 − y 2 / b 2 = 1 real points only when ∣ x ∣ ≥ a 2 a
Which branch is where x ≥ a x ≤ − a
Slope of the asymptotes a b
Meaning of e = c / a unitless shape score; e > 1
Meaning of "as ∣ x ∣ → ∞ watch the far-out behaviour where the "1
