On A Plane ...

Line

General Form: \( A x + B y + C = 0 \) Form 1: \( y = k x + b \)

Theorem 1

\( A x + B y + C \) have the same sign \( \iff \) Points are on the same side

\( A x + B y + C \) have the different sign \( \iff \) Points are on the different side

NCEE

Theorem 2

If the range of angles is continuous

90 degree is in the range and it is not the endpoint \( \iff \) the range of slope is discontinuous

90 degree is in the range \( \iff \) there is \( \infty \) in the expression of the range of slope

NCEE

NCEE: Parallel; Vertical and Auxiliary Rounding

Parametric Equation

Wikipedia

Procedure: reflect a point across a line

Given

Line \( Ax+By+C=0 \)

Point 0 \( P_0(x_0,y_0) \)

,

\(d_2=\frac{Ax_0+By_0+C}{A^2+B^2}\)

Point 1 \(x=x_0-2d_2A\) \(y=y_0-2d_2B\)

Circle

\((x-a)^2+(y-b)^2=r^2\)

Tangent:\((x-a)(x_0-a)+(y-b)(y_0-b)=r^2\)

Line AB: \((x-a)(x_0-a)+(y-b)(y_0-b)=r^2\)

NCEE

NCEE: Relations between multiple circles

Ellipse

Theorem:

\( S_{\Delta F_1 P F_2} = \frac{1}{2} \left\lvert F_1F_2 \right\rvert \left\lvert y_p \right\rvert = c \left\lvert y_p \right\rvert \)

\( S_{\Delta F_1 P F_2} = b^2 tan \frac{\theta}{2} \)

Theorem 1

\(c\in[sin\frac{\theta}2,1)\)

NCEE

Theorem 2

Ellipse/Hyperbola \(k_{PA}\cdot k_{PB}=e^2-1\)

Proof: \(x^2+(\frac{a}{b}y)^2=a^2\) Then regard it as a circle

(or, can alternatively be deduced from point difference method)

Ellipse/Hyperbola \(k_{MO}\cdot k_{BP}=e^2-1\)

Hyperbola

\( S_{\Delta F_1 P F_2} = b^2 cot \frac{\theta}{2} \)

NCEE

Parabola

Theorem 1

\(AB=AF+BF=\frac{2p}{1-cos^2\theta}=\frac{2p}{sin^2\theta}\)

Ellipse/Hyperbola: \(AF=\frac{\frac{b^2}a}{1-e\cdot cos\theta}BF=\frac{\frac{b^2}a}{1+e\cdot cos\theta}\)

Parabola/Ellipse/Hyperbola: \(AF=\lambda BF\iff \left\lvert e\cdot cos\theta\right\rvert=\left\lvert\frac{\lambda-1}{\lambda+1}\right\rvert\)

Mnemonic

\(\frac{p}{1\pm\left\lvert cos\theta\right\rvert}\) Long minus;short add

NCEE

设点联立韦达

1

Ellipse:

\(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) and \(Ax+By+C=0\)

\(\iff\)

\(\Delta=4b^2B^2a^2(a^2A^2+b^2B^2-C^2)\)

\(\Delta'=a^2A^2+b^2B^2-C^2\)

Hyperbola: \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\iff\frac{x^2}{a^2}+\frac{y^2}{(b i)^2}=1\) so \(\Delta'=C^2+b^2B^2-a^2A^2\)

NCEE

2

\(\alpha x^2+\beta y^2=\gamma\) and \(Ax^2+By+C=0\) \(\iff\) \((\alpha B^2+\beta A^2)x^2+2CA\beta x+\beta C^2-\gamma B^2\)

Ellipse: \(l=\frac{2\sqrt{a^2b^2(A^2+B^2)\Delta'}}{a^2A^2+b^2B^2}\)

Hyperbola: \(l=\frac{2\sqrt{a^2b^2(A^2+B^2)\Delta'}}{\left\lvert a^2A^2+b^2(Bi)^2 \right\rvert}=\frac{2\sqrt{a^2b^2(A^2+B^2)\Delta'}}{\left\lvert a^2A^2-b^2B^2 \right\rvert}\)