Whakaoti i te {l}{x^2/4+y^2/3=1}{y=k(x+1)}

$$\left\{ \begin{array} { l } { \frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 3 } = 1 } \\ { y = k ( x + 1 ) } \end{array} \right.$$

Whakaoti mō x, y

$$x=-\frac{2\left(2k^{2}+3\sqrt{k^{2}+1}\right)}{4k^{2}+3}\text{, }y=\frac{3k\left(-2\sqrt{k^{2}+1}+1\right)}{4k^{2}+3}$$

$$x=\frac{2\left(-2k^{2}+3\sqrt{k^{2}+1}\right)}{4k^{2}+3}\text{, }y=\frac{3k\left(2\sqrt{k^{2}+1}+1\right)}{4k^{2}+3}$$

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 12, arā, te tauraro pātahi he tino iti rawa te kitea o 4,3.

$$3x^{2}+4y^{2}=12$$

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te k ki te x+1.

$$y=kx+k$$

Whakakapia te kx+k mō te y ki tērā atu whārite, 3x^{2}+4y^{2}=12.

$$3x^{2}+4\left(kx+k\right)^{2}=12$$

Pūrua kx+k.

$$3x^{2}+4\left(k^{2}x^{2}+2kkx+k^{2}\right)=12$$

Whakareatia 4 ki te k^{2}x^{2}+2kkx+k^{2}.

$$3x^{2}+4k^{2}x^{2}+8k^{2}x+4k^{2}=12$$

Tāpiri 3x^{2} ki te 4k^{2}x^{2}.

$$\left(4k^{2}+3\right)x^{2}+8k^{2}x+4k^{2}=12$$

Me tango 12 mai i ngā taha e rua o te whārite.

$$\left(4k^{2}+3\right)x^{2}+8k^{2}x+4k^{2}-12=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 3+4k^{2} mō a, 4\times 2kk mō b, me 4k^{2}-12 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-8k^{2}±\sqrt{\left(8k^{2}\right)^{2}-4\left(4k^{2}+3\right)\left(4k^{2}-12\right)}}{2\left(4k^{2}+3\right)}$$

Pūrua 4\times 2kk.

$$x=\frac{-8k^{2}±\sqrt{64k^{4}-4\left(4k^{2}+3\right)\left(4k^{2}-12\right)}}{2\left(4k^{2}+3\right)}$$

Whakareatia -4 ki te 3+4k^{2}.

$$x=\frac{-8k^{2}±\sqrt{64k^{4}+\left(-16k^{2}-12\right)\left(4k^{2}-12\right)}}{2\left(4k^{2}+3\right)}$$

Whakareatia -12-16k^{2} ki te 4k^{2}-12.

$$x=\frac{-8k^{2}±\sqrt{64k^{4}+144+144k^{2}-64k^{4}}}{2\left(4k^{2}+3\right)}$$

Tāpiri 64k^{4} ki te 144+144k^{2}-64k^{4}.

$$x=\frac{-8k^{2}±\sqrt{144k^{2}+144}}{2\left(4k^{2}+3\right)}$$

Tuhia te pūtakerua o te 144k^{2}+144.

$$x=\frac{-8k^{2}±12\sqrt{k^{2}+1}}{2\left(4k^{2}+3\right)}$$

Whakareatia 2 ki te 3+4k^{2}.

$$x=\frac{-8k^{2}±12\sqrt{k^{2}+1}}{8k^{2}+6}$$

Nā, me whakaoti te whārite x=\frac{-8k^{2}±12\sqrt{k^{2}+1}}{8k^{2}+6} ina he tāpiri te ±. Tāpiri -8k^{2} ki te 12\sqrt{k^{2}+1}.

$$x=\frac{-8k^{2}+12\sqrt{k^{2}+1}}{8k^{2}+6}$$

Whakawehe -8k^{2}+12\sqrt{k^{2}+1} ki te 6+8k^{2}.

$$x=\frac{2\left(-2k^{2}+3\sqrt{k^{2}+1}\right)}{4k^{2}+3}$$

Nā, me whakaoti te whārite x=\frac{-8k^{2}±12\sqrt{k^{2}+1}}{8k^{2}+6} ina he tango te ±. Tango 12\sqrt{k^{2}+1} mai i -8k^{2}.

$$x=\frac{-8k^{2}-12\sqrt{k^{2}+1}}{8k^{2}+6}$$

Whakawehe -8k^{2}-12\sqrt{k^{2}+1} ki te 6+8k^{2}.

$$x=-\frac{2\left(2k^{2}+3\sqrt{k^{2}+1}\right)}{4k^{2}+3}$$

E rua ngā otinga mō x: \frac{2\left(-2k^{2}+3\sqrt{k^{2}+1}\right)}{3+4k^{2}} me -\frac{2\left(2k^{2}+3\sqrt{1+k^{2}}\right)}{3+4k^{2}}. Me whakakapi \frac{2\left(-2k^{2}+3\sqrt{k^{2}+1}\right)}{3+4k^{2}} mō x ki te whārite y=kx+k hei kimi i te otinga hāngai mō y e pai ai ki ngā whārite e rua.

$$y=k\times \frac{2\left(-2k^{2}+3\sqrt{k^{2}+1}\right)}{4k^{2}+3}+k$$

Whakareatia k ki te \frac{2\left(-2k^{2}+3\sqrt{k^{2}+1}\right)}{3+4k^{2}}.

$$y=\frac{2\left(-2k^{2}+3\sqrt{k^{2}+1}\right)}{4k^{2}+3}k+k$$

Me whakakapi te -\frac{2\left(2k^{2}+3\sqrt{1+k^{2}}\right)}{3+4k^{2}} ināianei mō te x ki te whārite y=kx+k ka whakaoti hei kimi i te otinga hāngai mō y e pai ai ki ngā whārite e rua.

$$y=k\left(-\frac{2\left(2k^{2}+3\sqrt{k^{2}+1}\right)}{4k^{2}+3}\right)+k$$

Whakareatia k ki te -\frac{2\left(2k^{2}+3\sqrt{1+k^{2}}\right)}{3+4k^{2}}.

$$y=\left(-\frac{2\left(2k^{2}+3\sqrt{k^{2}+1}\right)}{4k^{2}+3}\right)k+k$$

Kua oti te pūnaha te whakatau.

$$y=\frac{2\left(-2k^{2}+3\sqrt{k^{2}+1}\right)}{4k^{2}+3}k+k,x=\frac{2\left(-2k^{2}+3\sqrt{k^{2}+1}\right)}{4k^{2}+3}\text{ or }y=\left(-\frac{2\left(2k^{2}+3\sqrt{k^{2}+1}\right)}{4k^{2}+3}\right)k+k,x=-\frac{2\left(2k^{2}+3\sqrt{k^{2}+1}\right)}{4k^{2}+3}$$

Whakaoti mō x, y (complex solution)

$$\left\{\begin{matrix}x=-\frac{2\left(2k^{2}+3\sqrt{k^{2}+1}\right)}{4k^{2}+3}\text{, }y=\frac{3k\left(-2\sqrt{k^{2}+1}+1\right)}{4k^{2}+3}\text{; }x=\frac{2\left(-2k^{2}+3\sqrt{k^{2}+1}\right)}{4k^{2}+3}\text{, }y=\frac{3k\left(2\sqrt{k^{2}+1}+1\right)}{4k^{2}+3}\text{, }&k\neq -\frac{\sqrt{3}i}{2}\text{ and }k\neq \frac{\sqrt{3}i}{2}\\x=\frac{3-k^{2}}{2k^{2}}\text{, }y=\frac{k^{2}+3}{2k}\text{, }&k=-\frac{\sqrt{3}i}{2}\text{ or }k=\frac{\sqrt{3}i}{2}\end{matrix}\right.$$