Whakaoti i te {l}{-y_{1}(x_{1}+g)=(x-x_{1})(y_{1}+f).text{isa}}{text{Thepoint}P(x_{1},y_{1})text{liesoutside,on}}{y^2+2gx+2fy+c=0text{accordingas}}

$$\left. \begin{array} { l } { - y _ { 1 } ( x _ { 1 } + g ) = ( x - x _ { 1 } ) ( y _ { 1 } + f ) . \text { is a } } \\ { \text { The point } P ( x _ { 1 } , y _ { 1 } ) \text { lies outside, on } } \\ { y ^ { 2 } + 2 g x + 2 f y + c = 0 \text { according as } } \end{array} \right.$$

Whakaoti mō x

$$\left\{\begin{matrix}x=-\frac{gy_{1}-fx_{1}}{y_{1}+f}\text{, }&y_{1}\neq -f\\x\in \mathrm{R}\text{, }&\left(y_{1}=0\text{ and }f=0\right)\text{ or }\left(x_{1}=-g\text{ and }y_{1}=-f\right)\end{matrix}\right.$$

Ngā Upane mō te Whakaoti Whārite Rārangi

Whakamahia te āhuatanga tohatoha hei whakarea te -y_{1} ki te x_{1}+g.

$$\left(-y_{1}\right)x_{1}+\left(-y_{1}\right)g=\left(x-x_{1}\right)\left(y_{1}+f\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x-x_{1} ki te y_{1}+f.

$$\left(-y_{1}\right)x_{1}+\left(-y_{1}\right)g=xy_{1}+xf-x_{1}y_{1}-x_{1}f$$

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$xy_{1}+xf-x_{1}y_{1}-x_{1}f=\left(-y_{1}\right)x_{1}+\left(-y_{1}\right)g$$

Me tāpiri te x_{1}y_{1} ki ngā taha e rua.

$$xy_{1}+xf-x_{1}f=\left(-y_{1}\right)x_{1}+\left(-y_{1}\right)g+x_{1}y_{1}$$

Me tāpiri te x_{1}f ki ngā taha e rua.

$$xy_{1}+xf=\left(-y_{1}\right)x_{1}+\left(-y_{1}\right)g+x_{1}y_{1}+x_{1}f$$

Pahekotia te -y_{1}x_{1} me x_{1}y_{1}, ka 0.

$$xy_{1}+xf=-y_{1}g+x_{1}f$$

Pahekotia ngā kīanga tau katoa e whai ana i te x.

$$\left(y_{1}+f\right)x=-y_{1}g+x_{1}f$$

He hanga arowhānui tō te whārite.

$$\left(y_{1}+f\right)x=fx_{1}-gy_{1}$$

Whakawehea ngā taha e rua ki te y_{1}+f.

$$\frac{\left(y_{1}+f\right)x}{y_{1}+f}=\frac{fx_{1}-gy_{1}}{y_{1}+f}$$

Mā te whakawehe ki te y_{1}+f ka wetekia te whakareanga ki te y_{1}+f.

$$x=\frac{fx_{1}-gy_{1}}{y_{1}+f}$$