Before we start, one shared picture in words: stand at the centre of an ellipse. Walk right to the pointy end — that's the vertex, distance a (the semi-major axis). Walk up to the flat top — that's the co-vertex, distance b (the semi-minor axis). The two foci sit on the long axis, each a distance c from centre, and the master triangle a2=b2+c2 (legs b and c, hypotenuse a) ties them together — see Pythagorean theorem and the figure below. Keep that triangle in your head; almost every trap on this page is someone forgetting one of its three pieces.
The figure above shows both objects the traps rely on: the master triangle (legs b up to the co-vertex and c out to the focus, hypotenuse a to the vertex) and the latus rectum (the through-focus chord of height LR=2b2/a). Refer back to it whenever a card mentions "the triangle" or "the width at the focus."
The string 2a must be longer than the nail gap 2c, otherwise the pencil can't move; algebraically this keeps b2=a2−c2>0 so a real curve exists.
"Why do we call the constant sum 2a and not just k?"
Because that sum turns out to equal the full major-axis length, whose half is a; naming it 2a upfront makes the derived relations clean.
"Why does e=0 force a circle rather than just a rounder ellipse?"
e=0 means c=0, so b2=a2−0=a2, giving a=b — equal axes is exactly a circle, the only fully symmetric case.
"Why does the master triangle have hypotenuse a (not b or c)?"
Standing at the centre, the co-vertex is straight up (leg b) and the focus is straight out (leg c); the vertex distance a closes the right triangle as the longest side, so a2=b2+c2.
"Why does the larger denominator, not the larger coefficient, decide the major axis?"
In standard form each variable is divided by its denominator; a bigger denominator lets that variable reach a bigger value before the sum hits 1, so that direction is the long one.
"Why does the latus rectum formula come out as LR=2b2/a?"
Substitute the focus abscissa x=c into the ellipse: a2c2+b2y2=1 gives y2=b2a2a2−c2=a2b4, so y=±b2/a; the full chord is twice that, 2b2/a.
"Why is the latus rectum a useful 'size gauge' of the ellipse?"
It's the vertical width measured right at a focus, so it captures how fat the ellipse is at its most physically meaningful point without needing the full shape.
"What ellipse do you get as b→a (both semi-axes equal)?"
A Circle of radius a: c→0, foci merge at the centre, and e→0 — this is the round limit.
"What happens as b→0 (the minor axis collapses)?"
The oval flattens onto the segment joining (−a,0) and (a,0): here c=a2−b2→a and e→1, so the foci slide out to the vertices and the ellipse degenerates to a line segment — the flat limit.
"What happens to the shape as e→1− (but never reaching 1)?"
Equivalent to the b→0 story from the eccentricity side: the foci race toward the vertices, the ellipse becomes extremely elongated and thin, approaching that flat segment while staying closed until the Parabola limit.
"Can the two denominators be negative, e.g. −9x2+25y2=1?"
No ellipse: a negative denominator flips a sign and you'd get a Hyperbola — standard forms (or no real curve), since ellipses need both terms positive.
"For a2x2+a2y2=1 where are the foci and what is e?"
They coincide at the origin (c=a2−a2=0) and e=0 — this is the concrete circle instance of the b→a round limit.
"Is x2+4y2=0 an ellipse?"
No. The only real solution is the single point (0,0) (both squares must vanish), a fully degenerate case, not a curve.
Recall One-line survival kit
Defining property? ::: PF1+PF2=2a — the two focal distances add to the constant 2a for every point on the ellipse.
Which is a? ::: The larger semi-axis, sitting under the larger denominator and along the major axis.
Ellipse triangle? ::: a2=b2+c2 (subtract for c: c2=a2−b2).
Eccentricity, its formula and range? ::: e=c/a with 0≤e<1; bigger e = flatter, e=0 = circle, b→0 = flat segment limit.
