椭圆

# 定义

# 定义一

F1P+F2P=2aF1F2=2c |F_{1} P|+\left|F_{2} P\right|=2 a\\ \left|F_{1} F_{2}\right|=2 c\\
x2a2+y2b2=1(a>b>0) \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \quad(a>b>0)
PF1=a+exPF2=aex \begin{array}{l}{\left|P F_{1}\right|=a+e x} \\ {\left|P F_{2}\right|=a-e x}\end{array}\\

# 定义二

# 定义三

d=AA1F2Pd=e<1 d=|AA_1|\\ \frac{|F_2P|}{d}=e<1
CD=2b2a 通径 |CD|=2\frac{b^2}{a}
F2M=p=a2cc=b2c |F_2M|=p=\frac{a^2}{c}-c=\frac{b^2}{c}
AF2=ep1+ecosθ \left|A F_{2}\right|=\frac{e p}{1+e \cos \theta}\\
BF2=ep1ecosθ \left|B F_{2}\right|=\frac{e p}{1-e \cos \theta}\\
AB=2ep1e2cos2θ \left|AB\right|=\frac{2 e p}{1-e^2 \cos^2 \theta}\\
AF2BF2=1ecosθ1+ecosθ \frac{|AF_2|}{|BF_2|}=\frac{1-e\cos \theta}{1+e\cos \theta}

注意,通过控制 θ\theta 可以得出 ABmin=2b2a|AB|_{min}=2\frac{b^2}{a}

# 乱七八糟

# 三角换元

x2a2+y2b2=1(xa)2+(yb)2=1 \begin{aligned} \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}} &= 1 \\ \left(\frac{x}{a}\right)^{2}+\left(\frac{y}{b}\right)^{2} &= 1 \end{aligned}\\
{x=acosαy=bsinα {\left\{\begin{aligned} {x=a \cdot \cos \alpha} \\ {y=b \cdot \sin \alpha} \end{aligned}\right.}

# 点差法

{x12a2+y12b2=1x22a2+y22b2=1 \left\{\begin{aligned}{\frac{x_{1}^{2}}{a^{2}}+\frac{y_{1}^{2}}{b^{2}}=1} \\ {\frac{x_{2}^{2}}{a^{2}}+\frac{y_{2}^{2}}{b^{2}}=1}\end{aligned}\right.\\
x12x22a2=y12y02b2y1y2x1x2y1+y2x1+x2=b2a2y1y2x1x2y0x0=b2a2kABkOP=b2a2 \begin{aligned} \frac{x_{1}^{2}-x_{2}^{2}}{a^{2}}&=-\frac{y_{1}^{2}-y_{0}^{2}}{b^{2}} \\ \frac{y_{1}-y_{2}}{x_{1}-x_{2}} \cdot \frac{y_{1}+y_{2}}{x_{1}+x_{2}}&=-\frac{b^{2}}{a^{2}}\\ \frac{y_{1}-y_{2}}{x_{1}-x_{2}} \cdot \frac{y_0}{x_0}&=-\frac{b^{2}}{a^{2}}\\ k_{A B} \cdot k_{OP}&=-\frac{b^{2}}{a^{2}} \end{aligned}\\

# 切线

Ax2+By2+Cx+Dy+E=0Ax0x+By0y+Cx0+x2+Dy0+y2+E=0 \begin{gathered} A x^{2}+B y^{2}+C x+D y+E=0\\ A x_{0} x+B y_{0} y+C \frac{x_{0}+x}{2}+D \frac{y_{0}+y}{2}+E=0 \end{gathered}

# 整体思想

x1+kx2=0x1x2=kx2x1=1kx1x2+x2x1=x12+x22x1x2=(x1+x2)2x1x22(x1+x2)2x1x2=k1k+2 \begin{gathered} x_1 + kx_2 = 0\\ \frac{x_1}{x_2} = -k\\ \frac{x_2}{x_1} = -\frac{1}{k}\\ \frac{x_1}{x_2} + \frac{x_2}{x_1} = \frac{x_1^2 + x_2^2}{x_1x_2} = \frac{(x_1+x_2)^2}{x_1x_2}-2\\ \frac{(x_1+x_2)^2}{x_1x_2}=-k-\frac{1}{k}+2 \end{gathered}

# 焦点三角形

# 仿射变换

高中化学归纳

  • 反应条件

    • 光照
      • HClO\ce{HClO}
      • AgX+\ce{Ag+} (除 AgF\ce{AgF}
      • HNOX3\ce{HNO3}
      • HX2,ClX2\ce{H2, Cl2}
    • 放电
    • 高温
    • 催化剂,加热
      • HX2OX2\ce{H2O2}
      • NX2+HX2\ce{N2 + H2}
      • SOX2+OX2\ce{SO2 +O2}
      • MnOX2+KClOX3\ce{MnO2 + KClO3}
      • NHX3+OX2\ce{NH3 + O2}
  • 无机物性质

    • NHX3,HCl,HNOX3\ce{NH3,HCl,HNO3} 易挥发
    • MgX3NX2\ce{Mg3N2} 灰绿,M=100M=100
  • 实验

    • KMnOX4\ce{KMnO4} 腐蚀橡胶
    • NaOH\ce{NaOH} 腐蚀玻璃
    • 控制流速(气泡数),控制反应气比例(掌握仪器)

NHX3NX2NONOX2()HNOX3NHX3NOHNOX3 \ce{ NH3<=>N2<=>NO<=>NO2(红棕)<=>HNO3\\ NH3 -> NO <-HNO3 }

类比 S\ce{S},注意不可一步变成 NO\ce{NO}

NX2+OX2=2NO{2NO+OX2=2NOX23NOX2+HX2O=2HNOX3+NO{4NO+3OX2+HX2O=4HNOX34NOX2+OX2+2HX2O=4HNOX3 \ce{N2 + O2 = 2NO}\\ \left\{ \begin{aligned} \ce{2NO + O2 &= 2NO2}\\ \ce{3NO2 + H2O &= 2HNO3 + NO}\\ \end{aligned} \right. \rightarrow \left\{ \begin{aligned} \ce{4NO + 3O2 + H2O &= 4HNO3}\\ \ce{4NO2 + O2 + 2H2O &= 4HNO3}\\ \end{aligned} \right.

NHX3,HCl,HNOX3\ce{NH3,HCl,HNO3} 易挥发,空中反应,形成白烟

MgX3NX2\ce{Mg3N2} 灰绿,M=100M=100,双水解

注意 NX2+3HX22NHX3\ce{N2 + 3H2<=>2NH3} 转化率低,不会倒吸

尾气处理只要 NOX2 >NO\ce{NO2 >NO} 皆可除净(所以不行就通空气(OX2\ce{O2}))

2NOX2+2NaOHNaNOX3+NaNOX2+HX2O \ce{2NO2 + 2NaOH->NaNO3 + NaNO2 + H2O}
NOX2+NO+2NaOH2NaNOX2+HX2O \ce{NO2 + NO + 2NaOH->2NaNO2 + H2O}

#

氢键形式 HX3NHOH\ce{H3N\bond{...}H\bond{1}O\bond{1}H}

  • 氧化性 2Na+2NHX3=2NaNHX2+HX2\ce{2Na + 2NH3 = 2NaNH2 + H2}

  • 强还原

    4NHX3+5OX2ΔPt4NO+6HX2O \ce{4NH3 + 5O2->[Pt][\Delta]4NO + 6H2O}\\
    NHX3+3OX2()2NX2+6HX2O \ce{NH3 + 3O2(纯)->[点燃]2N2 + 6H2O}\\
    {2NHX3+3ClX2=NX2+6HCl8NHX3+3ClX2=NX2+6NHX4ClNHX3+3ClX2=NClX3+3HCl \left\{ \begin{aligned} \ce{2NH3 + 3Cl2&=N2 + 6HCl}\\ \ce{8NH3 + 3Cl2&=N2 + 6NH4Cl}\\ \ce{NH3 + 3Cl2&=NCl3 + 3HCl}\\ \end{aligned} \right.
  • 制取,平衡左移(OHX\ce{OH-}),放热

    CaO(s)+NHX3HX2O(l)Ca(OH)X2+NHX3 \ce{CaO(s) + NH3.H2O(l)->Ca(OH)2 + NH3 ^}\\
    NHX3HX2O(l)NaOH(s)NHX3+HX2O \ce{NH3.H2O(l)->[NaOH(s)]NH3 ^ + H2O}\\
    Ca(OH)X2+2NHX4Cl2NHX3+CaClX2 \ce{Ca(OH)2 + 2NH4Cl -> 2NH3 ^ + CaCl2}\\

控制流速(气泡数),控制反应气比例(掌握仪器)

# 氨盐

  • 白色固体
  • 离子晶体
  • 加热皆可分解
    • NHX3\ce{NH3 ^}
    • NHX4NOX3ΔNX2O+2HX2O\ce{NH4NO3 ->[\Delta]N2O + 2 H2O}
    • NHX4NOX2ΔNX2+2HX2O\ce{NH4NO2 ->[\Delta]N2 + 2H2O}

# 硝酸

  • 不稳定

    4HNOX3X/Δ4NOX2+OX2+2HX2O \ce{4HNO3_{浓} ->[光/\Delta] 4NO2 ^ + O2 ^ + 2H2O}
  • 酸性

    • 金属 不生成 HX2\ce{H2}
    • CuO,CaCOX3\ce{CuO, CaCO3} 就像普通酸
    • 紫色石蕊,先变红,后褪色
  • 强氧化

    • HNOX3X>HNOX3X>HX2SOX4X\ce{HNO3_{浓} > HNO3_{稀} > H2SO4_{浓}}

    • 金属

      • Pt,Au\ce{Pt, Au} 不反应

        • 和王水(HNOX3:HCl=1:3\ce{HNO3:HCl} = 1:3)反应
      • Fe,Cu\ce{Fe, Cu} 常温钝化

      • Cu\ce{Cu}

        Cu+4HNOX3XCu(NOX3)X2+2NOX2+4HX2O() \ce{Cu + 4HNO3_{浓} -> Cu(NO3)2 + 2NO2 ^ + 4H2O}(会稀释)
        3Cu+8HX++2NOX3X3CuX2++2NO+4HX2O \ce{3Cu + 8H+ + 2NO3- -> 3Cu^{2+} + 2NO ^ + 4H2O}
      • 非金属

        C+HNOX3X(Δ)4NOX2+COX2+2HX2O \ce{C + HNO3_{浓}->[(\Delta)] 4NO2 ^ + CO2 ^ + 2H2O}
        SX2,SOX3X2>IX>FeX2+>BrX >(ClX) \ce{S^{2-},SO3^{2-} > I- > Fe^{2+}>Br- >(Cl- 不会反应)}
  • 工业制取 (2233)

    1. NO\ce{NO} 生成 (转化器)

      4NHX3+5OX2ΔPt4NO+6HX2O \ce{4NH3 + 5O2 ->[Pt][\Delta] 4NO + 6H2O}
    2. HNOX3\ce{HNO3} 生成 (吸收塔)

      {2NO+OX2=2NOX23NOX2+HX2O=2HNOX3+NO4NO+3OX2+2HX2O=4HNOX3 \left\{ \begin{aligned} \ce{2NO + O2 &= 2NO2}\\ \ce{3NO2 + H2O &= 2HNO3 + NO}\\ \end{aligned} \right. \rightarrow \ce{4NO + 3O2 + 2H2O = 4HNO3}\\