Ellipse — standard forms, semi-major - minor axes, foci, eccentricity, latus rectum

1. WHAT is an ellipse (the defining property)

WHY the constant is called $2a$: we will see the constant equals the full length of the major axis, and the major axis has half-length $a$. Naming it $2a$ now saves algebra later.

2. HOW to derive the standard equation (from scratch)

Put the foci symmetrically on the $x$-axis: $F_1=(-c,0),\qquad F_2=(c,0),\qquad c>0.$

Let $P=(x,y)$. The defining property says: $\sqrt{(x+c)^2+y^2}+\sqrt{(x-c)^2+y^2}=2a.$

Step 1 — isolate one root. Why? Two square roots can't be squared away at once; isolate to kill one. $\sqrt{(x+c)^2+y^2}=2a-\sqrt{(x-c)^2+y^2}.$

Step 2 — square both sides. $(x+c)^2+y^2 = 4a^2 -4a\sqrt{(x-c)^2+y^2}+(x-c)^2+y^2.$

Step 3 — expand and cancel. Why? The $y^2$ and most $x$-terms cancel, leaving the surviving root alone. $x^2+2cx+c^2 = 4a^2 -4a\sqrt{(x-c)^2+y^2}+x^2-2cx+c^2.$ $4cx = 4a^2 - 4a\sqrt{(x-c)^2+y^2}.$ $a\sqrt{(x-c)^2+y^2} = a^2 - cx.$

Step 4 — square again. $a^2\big[(x-c)^2+y^2\big] = a^4 - 2a^2cx + c^2x^2.$ $a^2x^2 - 2a^2cx + a^2c^2 + a^2y^2 = a^4 - 2a^2cx + c^2x^2.$

Cancel $-2a^2cx$: $a^2x^2 + a^2c^2 + a^2y^2 = a^4 + c^2x^2.$ $(a^2-c^2)x^2 + a^2y^2 = a^2(a^2-c^2).$

Step 5 — define $b^2=a^2-c^2$. Why is this positive? Because $2a$ (the string) is longer than $2c$ (nail separation), so $a>c$, so $a^2-c^2>0$. Call it $b^2$: $b^2x^2 + a^2y^2 = a^2b^2.$

Divide by $a^2b^2$:

3. Reading the geometry off the equation

Set $y=0$: $x=\pm a$ → vertices $(\pm a,0)$. Set $x=0$: $y=\pm b$ → co-vertices $(0,\pm b)$.

Deriving latus rectum. Why do we care? It's the width of the ellipse across a focus — a quick "size" gauge. Put $x=c$ into the equation: $\frac{c^2}{a^2}+\frac{y^2}{b^2}=1 \Rightarrow y^2=b^2\left(1-\frac{c^2}{a^2}\right)=b^2\cdot\frac{a^2-c^2}{a^2}=\frac{b^2\cdot b^2}{a^2}=\frac{b^4}{a^2}.$ So $y=\pm b^2/a$, and the full chord length is:

4. The OTHER standard form (major axis vertical)

If the bigger denominator is under $y^2$: $\frac{x^2}{b^2}+\frac{y^2}{a^2}=1,\qquad a>b>0.$ Now the major axis is along the $y$-axis, foci at $(0,\pm c)$, $c=\sqrt{a^2-b^2}$.

5. Worked examples

Active-recall flashcards

Connections

• Conic Sections — overview (ellipse as a section of a cone with plane angle between base and slant)
• Circle (special case $e=0$, $a=b$)
• Hyperbola — standard forms (uses difference of focal distances; there $c^2=a^2+b^2$)
• Parabola ($e=1$ limiting boundary case)
• Eccentricity and directrix (focus–directrix unified definition $PF=e\cdot Pd$)
• Pythagorean theorem (the $a^2=b^2+c^2$ triangle)

Concept Map

Hinglish (regional understanding)