Whakaoti i te {l}{text{(4)}(acosalpha,2acosalpha)}{text{f}y+b=m_{1}(x+a)text{and}y+b=m_{2}(x+a)text{are}}{text{wotangentstotheparabola}y^2=4ax,text{then}}{{llll}{text{(1)}m_{1}+m_{2}=0}&{text{(2)}m_{1}m_{2}=1}}

$$\left. \begin{array} { l } { \text { (4) } ( a \cos \alpha , 2 a \cos \alpha ) } \\ { \text { f } y + b = m _ { 1 } ( x + a ) \text { and } y + b = m _ { 2 } ( x + a ) \text { are } } \\ { \text { wo tangents to the parabola } y ^ { 2 } = 4 a x , \text { then } } \\ { \left. \begin{array} { l l l l } { \text { (1) } m _ { 1 } + m _ { 2 } = 0 } & { \text { (2) } m _ { 1 } m _ { 2 } = 1 } \end{array} \right. } \end{array} \right.$$

Whakaoti mō x, y (complex solution)

$$\left\{\begin{matrix}\\x=-a\text{, }y=-b\text{, }&\text{unconditionally}\\x\in \mathrm{C}\text{, }y=-b\text{, }&m_{2}=0\text{ and }m_{1}=0\\x=\frac{y-am_{2}+b}{m_{2}}\text{, }y\in \mathrm{C}\text{, }&m_{2}\neq 0\text{ and }m_{1}=m_{2}\end{matrix}\right.$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Whakamahia te āhuatanga tohatoha hei whakarea te m_{1} ki te x+a.

$$y+b=m_{1}x+m_{1}a$$

Tangohia te m_{1}x mai i ngā taha e rua.

$$y+b-m_{1}x=m_{1}a$$

Tangohia te b mai i ngā taha e rua.

$$y-m_{1}x=m_{1}a-b$$

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te m_{2} ki te x+a.

$$y+b=m_{2}x+m_{2}a$$

Tangohia te m_{2}x mai i ngā taha e rua.

$$y+b-m_{2}x=m_{2}a$$

Tangohia te b mai i ngā taha e rua.

$$y-m_{2}x=m_{2}a-b$$

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.

$$y+\left(-m_{1}\right)x=am_{1}-b,y+\left(-m_{2}\right)x=am_{2}-b$$

Kōwhiria tētahi o ngā whārite ka whakaotia mō te y mā te wehe i te y i te taha mauī o te tohu ōrite.

$$y+\left(-m_{1}\right)x=am_{1}-b$$

Me tāpiri m_{1}x ki ngā taha e rua o te whārite.

$$y=m_{1}x+am_{1}-b$$

Whakakapia te m_{1}x+am_{1}-b mō te y ki tērā atu whārite, y+\left(-m_{2}\right)x=am_{2}-b.

$$m_{1}x+am_{1}-b+\left(-m_{2}\right)x=am_{2}-b$$

Tāpiri m_{1}x ki te -m_{2}x.

$$\left(m_{1}-m_{2}\right)x+am_{1}-b=am_{2}-b$$

Me tango am_{1}-b mai i ngā taha e rua o te whārite.

$$\left(m_{1}-m_{2}\right)x=a\left(m_{2}-m_{1}\right)$$

Whakawehea ngā taha e rua ki te m_{1}-m_{2}.

$$x=-a$$

Whakaurua te -a mō x ki y=m_{1}x+am_{1}-b. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=m_{1}\left(-a\right)+am_{1}-b$$

Whakareatia m_{1} ki te -a.

$$y=-am_{1}+am_{1}-b$$

Tāpiri am_{1}-b ki te -m_{1}a.

$$y=-b$$

Kua oti te pūnaha te whakatau.

$$y=-b,x=-a$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Whakamahia te āhuatanga tohatoha hei whakarea te m_{1} ki te x+a.

$$y+b=m_{1}x+m_{1}a$$

Tangohia te m_{1}x mai i ngā taha e rua.

$$y+b-m_{1}x=m_{1}a$$

Tangohia te b mai i ngā taha e rua.

$$y-m_{1}x=m_{1}a-b$$

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te m_{2} ki te x+a.

$$y+b=m_{2}x+m_{2}a$$

Tangohia te m_{2}x mai i ngā taha e rua.

$$y+b-m_{2}x=m_{2}a$$

Tangohia te b mai i ngā taha e rua.

$$y-m_{2}x=m_{2}a-b$$

Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.

$$y+\left(-m_{1}\right)x=am_{1}-b,y+\left(-m_{2}\right)x=am_{2}-b$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-m_{1}\\1&-m_{2}\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}am_{1}-b\\am_{2}-b\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&-m_{1}\\1&-m_{2}\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&-m_{1}\\1&-m_{2}\end{matrix}\right))\left(\begin{matrix}1&-m_{1}\\1&-m_{2}\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-m_{1}\\1&-m_{2}\end{matrix}\right))\left(\begin{matrix}am_{1}-b\\am_{2}-b\end{matrix}\right)$$

Ko te hua o tētahi poukapa me te kōaro ko te poukapa tuakiri.

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-m_{1}\\1&-m_{2}\end{matrix}\right))\left(\begin{matrix}am_{1}-b\\am_{2}-b\end{matrix}\right)$$

Whakareatia ngā poukapa kei te taha mauī o te tohu ōrite.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-m_{1}\\1&-m_{2}\end{matrix}\right))\left(\begin{matrix}am_{1}-b\\am_{2}-b\end{matrix}\right)$$

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right) te poukapa kōaro, nō reira ka taea te tuhi anō te whārite poukapa hei rapanga whakarea poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{m_{2}}{-m_{2}-\left(-m_{1}\right)}&-\frac{-m_{1}}{-m_{2}-\left(-m_{1}\right)}\\-\frac{1}{-m_{2}-\left(-m_{1}\right)}&\frac{1}{-m_{2}-\left(-m_{1}\right)}\end{matrix}\right)\left(\begin{matrix}am_{1}-b\\am_{2}-b\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{m_{2}}{m_{1}-m_{2}}&\frac{m_{1}}{m_{1}-m_{2}}\\-\frac{1}{m_{1}-m_{2}}&\frac{1}{m_{1}-m_{2}}\end{matrix}\right)\left(\begin{matrix}am_{1}-b\\am_{2}-b\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\left(-\frac{m_{2}}{m_{1}-m_{2}}\right)\left(am_{1}-b\right)+\frac{m_{1}}{m_{1}-m_{2}}\left(am_{2}-b\right)\\\left(-\frac{1}{m_{1}-m_{2}}\right)\left(am_{1}-b\right)+\frac{1}{m_{1}-m_{2}}\left(am_{2}-b\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-b\\-a\end{matrix}\right)$$

Tangohia ngā huānga poukapa y me x.

$$y=-b,x=-a$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Whakamahia te āhuatanga tohatoha hei whakarea te m_{1} ki te x+a.

$$y+b=m_{1}x+m_{1}a$$

Tangohia te m_{1}x mai i ngā taha e rua.

$$y+b-m_{1}x=m_{1}a$$

Tangohia te b mai i ngā taha e rua.

$$y-m_{1}x=m_{1}a-b$$

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te m_{2} ki te x+a.

$$y+b=m_{2}x+m_{2}a$$

Tangohia te m_{2}x mai i ngā taha e rua.

$$y+b-m_{2}x=m_{2}a$$

Tangohia te b mai i ngā taha e rua.

$$y-m_{2}x=m_{2}a-b$$

Hei whakaoti mā te tangohanga, ko ngā tau whakarea o tētahi o ngā taurangi me mātua ōrite i ngā whārite e rua kia whakakorehia ai te taurangi ina tangohia tētahi whārite mai i tētahi atu.

$$y+\left(-m_{1}\right)x=am_{1}-b,y+\left(-m_{2}\right)x=am_{2}-b$$

Me tango y+\left(-m_{2}\right)x=am_{2}-b mai i y+\left(-m_{1}\right)x=am_{1}-b mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$y-y+\left(-m_{1}\right)x+m_{2}x=am_{1}-b+b-am_{2}$$

Tāpiri y ki te -y. Ka whakakore atu ngā kupu y me -y, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$\left(-m_{1}\right)x+m_{2}x=am_{1}-b+b-am_{2}$$

Tāpiri -m_{1}x ki te m_{2}x.

$$\left(m_{2}-m_{1}\right)x=am_{1}-b+b-am_{2}$$

Tāpiri am_{1}-b ki te -m_{2}a+b.

$$\left(m_{2}-m_{1}\right)x=a\left(m_{1}-m_{2}\right)$$

Whakawehea ngā taha e rua ki te -m_{1}+m_{2}.

$$x=-a$$

Whakaurua te -a mō x ki y+\left(-m_{2}\right)x=am_{2}-b. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y+\left(-m_{2}\right)\left(-a\right)=am_{2}-b$$

Whakareatia -m_{2} ki te -a.

$$y+am_{2}=am_{2}-b$$

Me tango m_{2}a mai i ngā taha e rua o te whārite.

$$y=-b$$

Kua oti te pūnaha te whakatau.

$$y=-b,x=-a$$

Whakaoti mō x, y

$$\left\{\begin{matrix}\\x=-a\text{, }y=-b\text{, }&\text{unconditionally}\\x\in \mathrm{R}\text{, }y=-b\text{, }&m_{2}=0\text{ and }m_{1}=0\\x=\frac{y-am_{2}+b}{m_{2}}\text{, }y\in \mathrm{R}\text{, }&m_{2}\neq 0\text{ and }m_{1}=m_{2}\end{matrix}\right.$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Whakamahia te āhuatanga tohatoha hei whakarea te m_{1} ki te x+a.

$$y+b=m_{1}x+m_{1}a$$

Tangohia te m_{1}x mai i ngā taha e rua.

$$y+b-m_{1}x=m_{1}a$$

Tangohia te b mai i ngā taha e rua.

$$y-m_{1}x=m_{1}a-b$$

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te m_{2} ki te x+a.

$$y+b=m_{2}x+m_{2}a$$

Tangohia te m_{2}x mai i ngā taha e rua.

$$y+b-m_{2}x=m_{2}a$$

Tangohia te b mai i ngā taha e rua.

$$y-m_{2}x=m_{2}a-b$$

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.

$$y+\left(-m_{1}\right)x=am_{1}-b,y+\left(-m_{2}\right)x=am_{2}-b$$

Kōwhiria tētahi o ngā whārite ka whakaotia mō te y mā te wehe i te y i te taha mauī o te tohu ōrite.

$$y+\left(-m_{1}\right)x=am_{1}-b$$

Me tāpiri m_{1}x ki ngā taha e rua o te whārite.

$$y=m_{1}x+am_{1}-b$$

Whakakapia te m_{1}x+am_{1}-b mō te y ki tērā atu whārite, y+\left(-m_{2}\right)x=am_{2}-b.

$$m_{1}x+am_{1}-b+\left(-m_{2}\right)x=am_{2}-b$$

Tāpiri m_{1}x ki te -m_{2}x.

$$\left(m_{1}-m_{2}\right)x+am_{1}-b=am_{2}-b$$

Me tango am_{1}-b mai i ngā taha e rua o te whārite.

$$\left(m_{1}-m_{2}\right)x=a\left(m_{2}-m_{1}\right)$$

Whakawehea ngā taha e rua ki te m_{1}-m_{2}.

$$x=-a$$

Whakaurua te -a mō x ki y=m_{1}x+am_{1}-b. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=m_{1}\left(-a\right)+am_{1}-b$$

Whakareatia m_{1} ki te -a.

$$y=-am_{1}+am_{1}-b$$

Tāpiri am_{1}-b ki te -m_{1}a.

$$y=-b$$

Kua oti te pūnaha te whakatau.

$$y=-b,x=-a$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Whakamahia te āhuatanga tohatoha hei whakarea te m_{1} ki te x+a.

$$y+b=m_{1}x+m_{1}a$$

Tangohia te m_{1}x mai i ngā taha e rua.

$$y+b-m_{1}x=m_{1}a$$

Tangohia te b mai i ngā taha e rua.

$$y-m_{1}x=m_{1}a-b$$

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te m_{2} ki te x+a.

$$y+b=m_{2}x+m_{2}a$$

Tangohia te m_{2}x mai i ngā taha e rua.

$$y+b-m_{2}x=m_{2}a$$

Tangohia te b mai i ngā taha e rua.

$$y-m_{2}x=m_{2}a-b$$

Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.

$$y+\left(-m_{1}\right)x=am_{1}-b,y+\left(-m_{2}\right)x=am_{2}-b$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-m_{1}\\1&-m_{2}\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}am_{1}-b\\am_{2}-b\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&-m_{1}\\1&-m_{2}\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&-m_{1}\\1&-m_{2}\end{matrix}\right))\left(\begin{matrix}1&-m_{1}\\1&-m_{2}\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-m_{1}\\1&-m_{2}\end{matrix}\right))\left(\begin{matrix}am_{1}-b\\am_{2}-b\end{matrix}\right)$$

Ko te hua o tētahi poukapa me te kōaro ko te poukapa tuakiri.

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-m_{1}\\1&-m_{2}\end{matrix}\right))\left(\begin{matrix}am_{1}-b\\am_{2}-b\end{matrix}\right)$$

Whakareatia ngā poukapa kei te taha mauī o te tohu ōrite.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-m_{1}\\1&-m_{2}\end{matrix}\right))\left(\begin{matrix}am_{1}-b\\am_{2}-b\end{matrix}\right)$$

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right) te poukapa kōaro, nō reira ka taea te tuhi anō te whārite poukapa hei rapanga whakarea poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{m_{2}}{-m_{2}-\left(-m_{1}\right)}&-\frac{-m_{1}}{-m_{2}-\left(-m_{1}\right)}\\-\frac{1}{-m_{2}-\left(-m_{1}\right)}&\frac{1}{-m_{2}-\left(-m_{1}\right)}\end{matrix}\right)\left(\begin{matrix}am_{1}-b\\am_{2}-b\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{m_{2}}{m_{1}-m_{2}}&\frac{m_{1}}{m_{1}-m_{2}}\\-\frac{1}{m_{1}-m_{2}}&\frac{1}{m_{1}-m_{2}}\end{matrix}\right)\left(\begin{matrix}am_{1}-b\\am_{2}-b\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\left(-\frac{m_{2}}{m_{1}-m_{2}}\right)\left(am_{1}-b\right)+\frac{m_{1}}{m_{1}-m_{2}}\left(am_{2}-b\right)\\\left(-\frac{1}{m_{1}-m_{2}}\right)\left(am_{1}-b\right)+\frac{1}{m_{1}-m_{2}}\left(am_{2}-b\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-b\\-a\end{matrix}\right)$$

Tangohia ngā huānga poukapa y me x.

$$y=-b,x=-a$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Whakamahia te āhuatanga tohatoha hei whakarea te m_{1} ki te x+a.

$$y+b=m_{1}x+m_{1}a$$

Tangohia te m_{1}x mai i ngā taha e rua.

$$y+b-m_{1}x=m_{1}a$$

Tangohia te b mai i ngā taha e rua.

$$y-m_{1}x=m_{1}a-b$$

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te m_{2} ki te x+a.

$$y+b=m_{2}x+m_{2}a$$

Tangohia te m_{2}x mai i ngā taha e rua.

$$y+b-m_{2}x=m_{2}a$$

Tangohia te b mai i ngā taha e rua.

$$y-m_{2}x=m_{2}a-b$$

Hei whakaoti mā te tangohanga, ko ngā tau whakarea o tētahi o ngā taurangi me mātua ōrite i ngā whārite e rua kia whakakorehia ai te taurangi ina tangohia tētahi whārite mai i tētahi atu.

$$y+\left(-m_{1}\right)x=am_{1}-b,y+\left(-m_{2}\right)x=am_{2}-b$$

Me tango y+\left(-m_{2}\right)x=am_{2}-b mai i y+\left(-m_{1}\right)x=am_{1}-b mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$y-y+\left(-m_{1}\right)x+m_{2}x=am_{1}-b+b-am_{2}$$

Tāpiri y ki te -y. Ka whakakore atu ngā kupu y me -y, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$\left(-m_{1}\right)x+m_{2}x=am_{1}-b+b-am_{2}$$

Tāpiri -m_{1}x ki te m_{2}x.

$$\left(m_{2}-m_{1}\right)x=am_{1}-b+b-am_{2}$$

Tāpiri am_{1}-b ki te -m_{2}a+b.

$$\left(m_{2}-m_{1}\right)x=a\left(m_{1}-m_{2}\right)$$

Whakawehea ngā taha e rua ki te -m_{1}+m_{2}.

$$x=-a$$

Whakaurua te -a mō x ki y+\left(-m_{2}\right)x=am_{2}-b. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y+\left(-m_{2}\right)\left(-a\right)=am_{2}-b$$

Whakareatia -m_{2} ki te -a.

$$y+am_{2}=am_{2}-b$$

Me tango m_{2}a mai i ngā taha e rua o te whārite.

$$y=-b$$

Kua oti te pūnaha te whakatau.

$$y=-b,x=-a$$