CONICS

CONICS

1. (analytic definition) PF=ePD, where e is eccentricity. *在考纲里

   • conics can be defined as the loci of a point P moves so that PF=ePD. where F is one of the foci, e...(specified types of conics) and D lies on the corresponding directrix.

2. Parabola: PF=PD, e=1

   • general form → ①y=4ax^2, F(a,0), Ld: x=-a. ②x=4ay^2, F(0,a), Ld: y=-a.

3. Ellipse: PF=ePD, 0<e<1: e^2 = 1 - (b/a)^2

   • an ellipse can be defined as the locus of a point P which moves so that PF=ePD, where F is one of the foci, e&lt;1 and D lies on the corresponding directrix.
   • major axis and minor axis??
   • special facts → PF+PF'=2a
     • proof by definition: PF=ePD, PF'=ePD' → PF+PF' = e(PD+PD') = e(2a/e) = 2a.
     • proof by Pythagoros' theorem: use P(x, f(x)) OR P[parametric form]--> 简单不繁琐的 (根号里的式子可以factorize) ??
   • general form → ①(x/a)^2 + (y/b)^2 = 1

4. Hyperbola: PF=ePD, e>1: e^2 = 1+b^2/a^2)

   • where e is eccentricity, and 'a' 'b' are proportions. ??
   • special case → Rectangular hyperbola: asymtotes at right angle
     • 特殊性: 直角 e.g.tangent to E intercept two asymtotes therefore construct a normal angle triangle with two lines can easily be calculated with 1:1:√2 ??
     • general form → x^2 - y^2 = a^2 (a=b)
       • Asymtotes: y±x
   • general form → ①(x/a)^2 - (y/b)^2 = 1, F(±ae, o), Ld: x=±a/e
     • Asymtotes: y=±b/ax

5. PARAMETRIC EQUATION: reset equation in parametric representation, some parameters: time, angle...

   • Find it/ Find a cartesian locous equation
     • Mapping
       • in order to get a cartesian form (about 'x' and 'y') θ needed to be eliminate.&nbsp; 对于成立于f(θ) 的mapping, 里面的 '元素' 需要被f(x), f(y)替换. &lt;--难点在于思想: 找 '元素' 而非找 单独的 θ 或者 x,y.&nbsp;
       • e.g. about"1": sin plus cos, ±sech plus ±tanh. ±cosh minus ±sinh, ±sec minus ±tan, csc minus cot; about"-1": coth plus csch...
     • Domain
       • piecewise function
   • Parametric differentiation: dy/dx=(dy/dθ)/(dx/dθ)

6. FIND A CHORD EQUATION: two points needed in parametric form and use point-gradient function: (y-y1)/(x-x1)=m=(y1-y2)/(x1-x2)--> m can be simplified by identities *--> simplified/ final equation: constants at one side, varibles at the other side. *

   • (注意易错)'parallel chords'指任意两个chords mutually parallel, 而非chords parallel to ellipse or...
   • Equations of special chords
     • Chord of contacts = tangent at a given point
     • Focal chord

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