Foundations — Parametric forms of all conics

Before you can appreciate the parent topic, every symbol it throws at you must first be earned. Below, each idea is built from nothing, given a picture, and justified — in the exact order the topic needs them.

1. A point $(x,y)$ — the address on the plane

Why the topic needs it: a parametrization outputs $x=f(t)$ and $y=g(t)$ — two numbers glued into one address. If you don't see $(x,y)$ as "right-then-up", the phrase "the point traces the curve" is meaningless.

Negative numbers are allowed: $x<0$ means left, $y<0$ means down. This matters because conics live in all four quadrants (the four regions the axes cut the plane into):

• Quadrant I: $x>0,y>0$ (upper right)
• Quadrant II: $x<0,y>0$ (upper left)
• Quadrant III: $x<0,y<0$ (lower left)
• Quadrant IV: $x>0,y<0$ (lower right)

2. What "$x^2$" means as a picture

The whole parametric trick rests on one fact: $x^2+y^2$ and $x^2-y^2$ have fixed geometric meanings.

3. The right triangle — where angles are born

Why the topic needs it: the parameter $\theta$ in $(r\cos \theta ,r\sin \theta )$ is an angle in exactly such a triangle. To read "the point at angle $\theta$", you must picture that triangle.

4. sin, cos, tan, sec — ratios that encode an angle

Why a ratio and not the raw length? A ratio cancels the size of the triangle. Two triangles of different scale but the same angle give the same $\sin ,\cos ,\tan$. So these numbers encode the angle alone — exactly what we want to feed as our single dial.

The unit circle picture makes them cleaner still: put the hypotenuse to length $1$ and stand it at angle $\theta$ from the positive $x$-axis. Then the tip of that radius sits at $(\cos \theta ,\sin \theta )$ — no fractions at all.

For the full toolkit of relations between these, see Trigonometric identities.

5. The identities — the engines that copy the conic's shape

The first identity is Pythagoras on the unit circle (side${}^2$ + side${}^2$ = hypotenuse${}^2$ = $1$). The second is that same identity divided through by ${\cos }^2\theta$ — try it: $\frac{{\cos }^2}{{\cos }^2}+\frac{{\sin }^2}{{\cos }^2}=\frac{1}{{\cos }^2}$ gives $1+{\tan }^2={\sec }^2$, i.e. ${\sec }^2-{\tan }^2=1$.

6. Radians and the range of $\theta$

Why the topic needs it: the parent lists ranges like $\theta \in [0,2\pi )$ (one full lap, no repeat) and $\theta \ne \frac{\pi }{2}$ (where $\sec \theta$ blows up). Those brackets are radian statements. $[0,2\pi )$ means "from $0$ up to but not including $2\pi$" — because $2\pi$ would land on the same point as $0$.

7. sinh, cosh — the hyperbola's other engine

Why a second engine for the same conic? The trig form $(a\sec \theta ,b\tan \theta )$ reaches both branches of the hyperbola but has an awkward blow-up at $\theta =\frac{\pi }{2}$. The hyperbolic form is smooth for every real $t$ but $\cosh t\ge 1$, so it only reaches the right branch. Different tools, different coverage — the parent's "misses a branch" mistake lives here. Details in Hyperbolic functions.

8. Slope and parametric differentiation (a peek)

When a curve is given by $x=f(t),y=g(t)$, its slope is $\frac{dy}{dx}=\frac{dy/dt}{dx/dt},$ where $\frac{dy}{dt}$ is "how fast $y$ changes as $t$ ticks". This is why tangents to conics become one-line algebra in $t$ (as in the parent's parabola tangent $ty=x+a{t}^2$). Full machinery: Parametric differentiation.

How these feed the topic

Each arrow is a "you cannot skip me": no ratios without a triangle, no identity without ratios, no parametrization without the identity.

Equipment checklist

Self-test: can you answer each before revealing?

Connections

• Parametric forms of all conics
• Conic Sections - Standard equations
• Trigonometric identities
• Hyperbolic functions
• Parametric differentiation
• Eccentric angle and auxiliary circle