the-circles-x-y-4x-2y-3-0-and-x-y-2x-4y-3-0-touches-each-other-Find-the-coordinates-of-the-point-of-contact-

Question Number 216630 by Tawa11 last updated on 12/Feb/25

the circles x² + y² -4x -2y +3 =0 and
x² +y² + 2x +4y -3 =0 touches each other
Find the coordinates of the point of contact

Answered by A5T last updated on 12/Feb/25

$$\left(\mathrm{ii}\right)−\left(\mathrm{i}\right)\Rightarrow\mathrm{6x}+\mathrm{6y}=\mathrm{6}\Rightarrow\mathrm{x}+\mathrm{y}=\mathrm{1} \\ $$$$\Rightarrow\left(\mathrm{1}−\mathrm{y}\right)^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} −\mathrm{4}\left(\mathrm{1}−\mathrm{y}\right)−\mathrm{2y}+\mathrm{3}=\mathrm{0} \\ $$$$\Rightarrow\mathrm{y}=\mathrm{0}\Rightarrow\:\mathrm{x}=\mathrm{1} \\ $$$$\Rightarrow\mathrm{The}\:\mathrm{circles}\:\mathrm{touch}\:\mathrm{each}\:\mathrm{other}\:\mathrm{at}\:\left(\mathrm{1},\mathrm{0}\right) \\ $$

Commented by Tawa11 last updated on 13/Feb/25

$$\mathrm{Thanks}\:\mathrm{sir},\:\mathrm{I}\:\mathrm{appreciate}. \\ $$

Answered by mr W last updated on 12/Feb/25

$$\left({x}−\mathrm{2}\right)^{\mathrm{2}} +\left({y}−\mathrm{1}\right)^{\mathrm{2}} =\left(\sqrt{\mathrm{2}}\right)^{\mathrm{2}} \\ $$$$\left({x}+\mathrm{1}\right)^{\mathrm{2}} +\left({y}+\mathrm{2}\right)^{\mathrm{2}} =\left(\mathrm{2}\sqrt{\mathrm{2}}\right)^{\mathrm{2}} \\ $$$${touching}\:{point}: \\ $$$$\left(\mathrm{2}+\frac{\sqrt{\mathrm{2}}}{\mathrm{3}\sqrt{\mathrm{2}}}×\left(−\mathrm{1}−\mathrm{2}\right),\:\mathrm{1}+\frac{\sqrt{\mathrm{2}}}{\mathrm{3}\sqrt{\mathrm{2}}}×\left(−\mathrm{2}−\mathrm{1}\right)\right) \\ $$$$\Rightarrow\left(\mathrm{1},\:\mathrm{0}\right) \\ $$

Commented by mr W last updated on 13/Feb/25

Commented by mr W last updated on 13/Feb/25

$${x}_{{c}} ={x}_{\mathrm{1}} +\frac{{r}_{\mathrm{1}} }{{r}_{\mathrm{1}} +{r}_{\mathrm{2}} }\left({x}_{\mathrm{2}} −{x}_{\mathrm{1}} \right) \\ $$$${y}_{{c}} ={y}_{\mathrm{1}} +\frac{{r}_{\mathrm{1}} }{{r}_{\mathrm{1}} +{r}_{\mathrm{2}} }\left({y}_{\mathrm{2}} −{y}_{\mathrm{1}} \right) \\ $$

Commented by Tawa11 last updated on 13/Feb/25

$$\mathrm{Thanks}\:\mathrm{sir}. \\ $$$$\mathrm{I}\:\mathrm{appreciate}. \\ $$

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