Whakaoti i te {l}{x^2/4+y^2/2=1}{x=my+1}

$$\left\{ \begin{array} { l } { \frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 2 } = 1 } \\ { x = m y + 1 } \end{array} \right.$$

Whakaoti mō x, y

$$x=\frac{\sqrt{2}\left(-m\sqrt{2m^{2}+3}+\sqrt{2}\right)}{m^{2}+2}\text{, }y=-\frac{\sqrt{2\left(2m^{2}+3\right)}+m}{m^{2}+2}$$

$$x=\frac{\sqrt{2}\left(m\sqrt{2m^{2}+3}+\sqrt{2}\right)}{m^{2}+2}\text{, }y=\frac{\sqrt{2\left(2m^{2}+3\right)}-m}{m^{2}+2}$$

Ngā Upane e Whakamahi ana i te Whakakapinga me te Tikanga Tātai Pūrua

Whakaarohia te whārite tuatahi. Me whakarea ngā taha e rua o te whārite ki te 4, arā, te tauraro pātahi he tino iti rawa te kitea o 4,2.

$$x^{2}+2y^{2}=4$$

Whakaarohia te whārite tuarua. Tangohia te my mai i ngā taha e rua.

$$x-my=1$$

Hei whakaoti i ētahi whārite takirua mā te whakakapinga, me whakaoti tētahi whārite i te tuatahi mō tētahi o ngā taurangi. Ka whakakapi i te otinga mō taua taurangi ki tērā o ngā whārite.

$$x+\left(-m\right)y=1,2y^{2}+x^{2}=4$$

Whakaotia te x+\left(-m\right)y=1 mō x mā te wehe i te x i te taha mauī o te tohu ōrite.

$$x+\left(-m\right)y=1$$

Me tango \left(-m\right)y mai i ngā taha e rua o te whārite.

$$x=my+1$$

Whakakapia te my+1 mō te x ki tērā atu whārite, 2y^{2}+x^{2}=4.

$$2y^{2}+\left(my+1\right)^{2}=4$$

Pūrua my+1.

$$2y^{2}+m^{2}y^{2}+2my+1=4$$

Tāpiri 2y^{2} ki te m^{2}y^{2}.

$$\left(m^{2}+2\right)y^{2}+2my+1=4$$

Me tango 4 mai i ngā taha e rua o te whārite.

$$\left(m^{2}+2\right)y^{2}+2my-3=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 2+1m^{2} mō a, 1\times 1\times 2m mō b, me -3 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-2m±\sqrt{\left(2m\right)^{2}-4\left(m^{2}+2\right)\left(-3\right)}}{2\left(m^{2}+2\right)}$$

Pūrua 1\times 1\times 2m.

$$y=\frac{-2m±\sqrt{4m^{2}-4\left(m^{2}+2\right)\left(-3\right)}}{2\left(m^{2}+2\right)}$$

Whakareatia -4 ki te 2+1m^{2}.

$$y=\frac{-2m±\sqrt{4m^{2}+\left(-4m^{2}-8\right)\left(-3\right)}}{2\left(m^{2}+2\right)}$$

Whakareatia -8-4m^{2} ki te -3.

$$y=\frac{-2m±\sqrt{4m^{2}+12m^{2}+24}}{2\left(m^{2}+2\right)}$$

Tāpiri 4m^{2} ki te 24+12m^{2}.

$$y=\frac{-2m±\sqrt{16m^{2}+24}}{2\left(m^{2}+2\right)}$$

Tuhia te pūtakerua o te 24+16m^{2}.

$$y=\frac{-2m±2\sqrt{4m^{2}+6}}{2\left(m^{2}+2\right)}$$

Whakareatia 2 ki te 2+1m^{2}.

$$y=\frac{-2m±2\sqrt{4m^{2}+6}}{2m^{2}+4}$$

Nā, me whakaoti te whārite y=\frac{-2m±2\sqrt{4m^{2}+6}}{2m^{2}+4} ina he tāpiri te ±. Tāpiri -2m ki te 2\sqrt{6+4m^{2}}.

$$y=\frac{2\sqrt{4m^{2}+6}-2m}{2m^{2}+4}$$

Whakawehe -2m+2\sqrt{6+4m^{2}} ki te 4+2m^{2}.

$$y=\frac{\sqrt{4m^{2}+6}-m}{m^{2}+2}$$

Nā, me whakaoti te whārite y=\frac{-2m±2\sqrt{4m^{2}+6}}{2m^{2}+4} ina he tango te ±. Tango 2\sqrt{6+4m^{2}} mai i -2m.

$$y=\frac{-2\sqrt{4m^{2}+6}-2m}{2m^{2}+4}$$

Whakawehe -2m-2\sqrt{6+4m^{2}} ki te 4+2m^{2}.

$$y=-\frac{\sqrt{4m^{2}+6}+m}{m^{2}+2}$$

E rua ngā otinga mō y: \frac{-m+\sqrt{6+4m^{2}}}{2+m^{2}} me -\frac{m+\sqrt{6+4m^{2}}}{2+m^{2}}. Me whakakapi \frac{-m+\sqrt{6+4m^{2}}}{2+m^{2}} mō y ki te whārite x=my+1 hei kimi i te otinga hāngai mō x e pai ai ki ngā whārite e rua.

$$x=m\times \frac{\sqrt{4m^{2}+6}-m}{m^{2}+2}+1$$

Whakareatia m ki te \frac{-m+\sqrt{6+4m^{2}}}{2+m^{2}}.

$$x=\frac{\sqrt{4m^{2}+6}-m}{m^{2}+2}m+1$$

Tāpiri m\times \frac{-m+\sqrt{6+4m^{2}}}{2+m^{2}} ki te 1.

$$x=1+\frac{\sqrt{4m^{2}+6}-m}{m^{2}+2}m$$

Me whakakapi te -\frac{m+\sqrt{6+4m^{2}}}{2+m^{2}} ināianei mō te y ki te whārite x=my+1 ka whakaoti hei kimi i te otinga hāngai mō x e pai ai ki ngā whārite e rua.

$$x=m\left(-\frac{\sqrt{4m^{2}+6}+m}{m^{2}+2}\right)+1$$

Whakareatia m ki te -\frac{m+\sqrt{6+4m^{2}}}{2+m^{2}}.

$$x=\left(-\frac{\sqrt{4m^{2}+6}+m}{m^{2}+2}\right)m+1$$

Tāpiri m\left(-\frac{m+\sqrt{6+4m^{2}}}{2+m^{2}}\right) ki te 1.

$$x=1+\left(-\frac{\sqrt{4m^{2}+6}+m}{m^{2}+2}\right)m$$

Kua oti te pūnaha te whakatau.

$$x=1+\frac{\sqrt{4m^{2}+6}-m}{m^{2}+2}m,y=\frac{\sqrt{4m^{2}+6}-m}{m^{2}+2}\text{ or }x=1+\left(-\frac{\sqrt{4m^{2}+6}+m}{m^{2}+2}\right)m,y=-\frac{\sqrt{4m^{2}+6}+m}{m^{2}+2}$$

Whakaoti mō x, y (complex solution)

$$\left\{\begin{matrix}x=\frac{\sqrt{2}\left(-m\sqrt{2m^{2}+3}+\sqrt{2}\right)}{m^{2}+2}\text{, }y=-\frac{\sqrt{2\left(2m^{2}+3\right)}+m}{m^{2}+2}\text{; }x=\frac{\sqrt{2}\left(m\sqrt{2m^{2}+3}+\sqrt{2}\right)}{m^{2}+2}\text{, }y=\frac{\sqrt{2\left(2m^{2}+3\right)}-m}{m^{2}+2}\text{, }&m\neq -\sqrt{2}i\text{ and }m\neq \sqrt{2}i\\x=\frac{5}{2}=2.5\text{, }y=\frac{3}{2m}\text{, }&m=-\sqrt{2}i\text{ or }m=\sqrt{2}i\end{matrix}\right.$$