Intuition The ONE core ideaAn ellipse is the set of all points whose two distances to two fixed pins add up to the same number every time . That fixed number is called 2 a Sum of Focal Radii = 2a .
Before you can derive that "constant sum," you need to be fluent in a small pile of symbols and pictures that the parent note quietly assumes. This page builds every one of them from nothing, in an order where each idea rests on the previous. Read top to bottom.
A point is a single dot on a flat sheet. To pin down which dot, we give it two numbers: how far right (x up (y origin O P = ( x , y )
Picture: a flat cream sheet with two number-lines crossing at the middle. Slide x y P
Why the topic needs this. The whole ellipse is described as "the collection of points ( x , y )
x left ) and y down ). Keep that in mind — signs matter later.
Suppose two dots A = ( x 1 , y 1 ) B = ( x 2 , y 2 )
Intuition Why a square root — and whythis tool
The straight gap between two dots is the hypotenuse (the slanted long side) of a right-angled triangle whose flat legs are the sideways gap x 2 − x 1 y 2 − y 1 Pythagoras theorem — "leg² + leg² = hypotenuse²." To undo that square and recover the length itself, we take a square root. That is the only reason
Look at the figure: the burnt-orange slanted line is A B − 3 + 3 ≥ 0
Why the topic needs this. A "focal radius" is literally a distance from a point to a focus. Step 1 of the parent's derivation is this formula. Everything downstream is algebra on this one square root.
a 2 a oval . Its longest straight width across the middle is the major axis ; the parent calls its full length 2 a vertex ) — is a semi-major axis .
Picture: the oval's widest cut, drawn burnt-orange. Centre to the right tip is a 2 a
Why the topic needs this. 2 a is the hidden constant — the fixed number the two distances add up to. The entire property is the sentence "r 1 + r 2 = 2 a a
b shortest width across the middle (perpendicular to the major axis) is the minor axis ; from centre to the top edge is b semi-minor axis . For a genuine oval we insist a > b > 0
In the same figure above, the teal vertical half-height is b a = b
Why the topic needs this. b 2 a bridge that lets us erase y b
Definition Foci and focal radiiInside the oval sit two special pinned points called foci (one focus, two foci). We name them S S ′ P r 1 = S P ( distance to S ) , r 2 = S ′ P ( distance to S ′ ) focal radii of P
Picture the gardener's trick : two nails (S S ′ r 1 r 2 sum is frozen at the string's length. That frozen length is exactly 2 a
Why the topic needs this. These are the two things we add. The whole property compares r 1 + r 2 r 1 − r 2 Definition of Conic as Locus for the "constant sum defines the curve" viewpoint.
e Eccentricity e how squashed the oval is. For an ellipse 0 < e < 1 e = 0 e 1
The foci live at x = ± a e e a e
Why this relation, and why the topic needs it. In the derivation we must kill y 2 . The ellipse equation lets us swap y 2 b 2 b 2 a e b 2 a e exactly what makes the mess fold into a perfect square. It's the hinge of the whole proof. More on e Eccentricity of Conics .
Read this as a membership test : plug a point ( x , y ) 1 on the oval. This is where a b x y y 2 = b 2 ( 1 − x 2 / a 2 ) y Ellipse - Standard Equation .
Definition Perfect squareA perfect square is an expression that is literally "something times itself," like ( a − e x ) 2 ( something ) 2 ∣ something ∣ becomes a perfect square, letting the ugly root vanish cleanly.
Definition Absolute value∣ q ∣ q ∣ − 4 ∣ = 4 ∣4∣ = 4 q 2 = ∣ q ∣ not just q
( a − e x ) 2 a − e x Why it feels right: square and root look like they simply cancel. Fix: in general ( a − e x ) 2 = ∣ a − e x ∣ on the ellipse − a ≤ x ≤ a 0 < e < 1 a − e x > 0 ∣ a − e x ∣ = a − e x
The Directrix and Focal Distance view gives another route to a − e x
Coordinates x y of a point P
Distance formula square root
Focal radii r1 = SP and r2 = S'P
Semi major axis a gives 2a
Eccentricity e places foci at plus minus ae
Bridge b squared = a squared times 1 minus e squared
Replace y squared using the equation
Perfect square a minus ex squared
Absolute value and sign check
Take square root to get a minus ex
Sum of focal radii equals 2a property
Reveal each after you've answered it out loud.
What does P = ( x , y ) x A point at x y x
Write the distance between ( x 1 , y 1 ) ( x 2 , y 2 ) Why does a square root appear in distance? Pythagoras gives leg² + leg² = length²; the root undoes the square to recover the length.
What is 2 a The full length of the major axis — the oval's longest width.
What is a b a b a > b > 0
What are S S ′ r 1 r 2 The two foci and the two distances from a point P
Where do the foci sit and what controls that? At x = ± a e e 0 < e < 1
State the bridge relation and why we need it. b 2 = a 2 ( 1 − e 2 ) b 2 a , e
What does the ellipse equation test? Whether a point is on the oval — plug in ( x , y ) 1
Why is ( a − e x ) 2 = a − e x Because
q 2 = ∣ q ∣ , and here
a − e x > 0 for points on the ellipse, so the bars drop safely.
