The equations of two circles S_1 and S_2 are given by S_1 : x^2 + y^2 +2x +2y + 1 = 0 S_2 : x^2 + y^2 −4x + 2y +1 = 0. Show that S_1 and S_2 touch each other externally and obtain the equation of the common tangent T at the point of contact.


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Question Number 96340 by Rio Michael last updated on 31/May/20

$The equations of two circles S_{1} and S_{2} are given by$ $S_{1} : x^{2} + y^{2} + 2 x + 2 y + 1 = 0$ $S_{2} : x^{2} + y^{2} - 4 x + 2 y + 1 = 0 .$ $Show that S_{1} and S_{2} touch each other externally and obtain$ $the equation of the common tangent T at the point of contact .$
Commented by PRITHWISH SEN 2 last updated on 31/May/20

$The distance between the centers = The sum of$ $their radius$ $Now prove it . I know you can .$
Commented by Rio Michael last updated on 31/May/20

$so you mean C_{1} C_{2} = r_{1} + r_{2}$ $S_{1} : {(x + 1)}^{2} + {(y + 1)}^{2} = 1^{2} \Rightarrow C_{1} (- 1, - 1), r_{1} = 1$ $S_{2} : {(x - 2)}^{2} + (y + 1) = 2^{2} \Rightarrow C_{1} (2, - 1), r_{2} = 2$ $C_{1} C_{2} = \sqrt{3^{2}} = 3 units .$ $r_{1} + r_{2} = 1 + 2 = 3 units$ $\Rightarrow they touch each other externally .$ $sir whats the idea for the second part ?$
Commented by PRITHWISH SEN 2 last updated on 31/May/20

$Find the common point by solving the two$ $equations . And then get the slope of the line joining$ $that point with any one center .$ $Then keep in mind that the radius and the tangent$ $at the common point are ⊥ to each other .$ $So find the slope of the tangent . And then . . . . .$ $you know what to do .$
Commented by Rio Michael last updated on 01/Jun/20

$thank you sir .$
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