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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 +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.
$$\mathrm{The}\:\mathrm{equations}\:\mathrm{of}\:\mathrm{two}\:\mathrm{circles}\:{S}_{\mathrm{1}} \:\mathrm{and}\:{S}_{\mathrm{2}} \:\mathrm{are}\:\mathrm{given}\:\mathrm{by} \\ $$$$\:{S}_{\mathrm{1}} :\:{x}^{\mathrm{2}} \:+\:{y}^{\mathrm{2}} \:+\mathrm{2}{x}\:+\mathrm{2}{y}\:+\:\mathrm{1}\:=\:\mathrm{0} \\ $$$$\:\:\:{S}_{\mathrm{2}} :\:{x}^{\mathrm{2}} \:+\:{y}^{\mathrm{2}} \:−\mathrm{4}{x}\:+\:\mathrm{2}{y}\:+\mathrm{1}\:=\:\mathrm{0}. \\ $$$$\mathrm{Show}\:\mathrm{that}\:{S}_{\mathrm{1}} \:\mathrm{and}\:{S}_{\mathrm{2}} \:\mathrm{touch}\:\mathrm{each}\:\mathrm{other}\:\mathrm{externally}\:\mathrm{and}\:\mathrm{obtain} \\ $$$$\mathrm{the}\:\mathrm{equation}\:\mathrm{of}\:\mathrm{the}\:\mathrm{common}\:\mathrm{tangent}\:{T}\:\mathrm{at}\:\mathrm{the}\:\mathrm{point}\:\mathrm{of}\:\mathrm{contact}. \\ $$
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.
$$\mathrm{The}\:\mathrm{distance}\:\mathrm{between}\:\mathrm{the}\:\mathrm{centers}\:=\:\mathrm{The}\:\mathrm{sum}\:\mathrm{of} \\ $$$$\mathrm{their}\:\mathrm{radius} \\ $$$$\mathrm{Now}\:\mathrm{prove}\:\mathrm{it}.\:\mathrm{I}\:\mathrm{know}\:\mathrm{you}\:\mathrm{can}. \\ $$
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 ⇒ C_1 (−1,−1) , r_1 = 1 S_2 : (x−2)^2 + (y + 1) = 2^2 ⇒ C_1 (2,−1), r_2 = 2 C_1 C_2 = (√3^2 ) = 3 units. r_1 + r_2 = 1 +2 = 3 units ⇒ they touch each other externally. sir whats the idea for the second part?
$$\mathrm{so}\:\mathrm{you}\:\mathrm{mean}\:{C}_{\mathrm{1}} {C}_{\mathrm{2}} \:=\:{r}_{\mathrm{1}} \:+\:{r}_{\mathrm{2}} \: \\ $$$${S}_{\mathrm{1}} :\:\left({x}\:+\:\mathrm{1}\right)^{\mathrm{2}} \:\:+\left({y}\:+\:\mathrm{1}\right)^{\mathrm{2}} \:=\:\mathrm{1}^{\mathrm{2}} \:\Rightarrow\:{C}_{\mathrm{1}} \left(−\mathrm{1},−\mathrm{1}\right)\:,\:{r}_{\mathrm{1}} \:=\:\mathrm{1} \\ $$$${S}_{\mathrm{2}} :\:\left({x}−\mathrm{2}\right)^{\mathrm{2}} +\:\left({y}\:+\:\mathrm{1}\right)\:=\:\mathrm{2}^{\mathrm{2}} \:\Rightarrow\:{C}_{\mathrm{1}} \left(\mathrm{2},−\mathrm{1}\right),\:{r}_{\mathrm{2}} \:=\:\mathrm{2} \\ $$$${C}_{\mathrm{1}} {C}_{\mathrm{2}} \:=\:\sqrt{\mathrm{3}^{\mathrm{2}} }\:=\:\mathrm{3}\:\mathrm{units}. \\ $$$$\:{r}_{\mathrm{1}} \:+\:{r}_{\mathrm{2}} \:=\:\mathrm{1}\:+\mathrm{2}\:=\:\mathrm{3}\:\mathrm{units} \\ $$$$\Rightarrow\:\mathrm{they}\:\mathrm{touch}\:\mathrm{each}\:\mathrm{other}\:\mathrm{externally}. \\ $$$$\mathrm{sir}\:\mathrm{whats}\:\mathrm{the}\:\mathrm{idea}\:\mathrm{for}\:\mathrm{the}\:\mathrm{second}\:\mathrm{part}? \\ $$
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.
$$\mathrm{Find}\:\mathrm{the}\:\mathrm{common}\:\mathrm{point}\:\mathrm{by}\:\mathrm{solving}\:\mathrm{the}\:\mathrm{two}\: \\ $$$$\mathrm{equations}.\:\mathrm{And}\:\mathrm{then}\:\mathrm{get}\:\mathrm{the}\:\mathrm{slope}\:\mathrm{of}\:\mathrm{the}\:\mathrm{line}\:\mathrm{joining} \\ $$$$\mathrm{that}\:\mathrm{point}\:\mathrm{with}\:\mathrm{any}\:\mathrm{one}\:\mathrm{center}. \\ $$$$\mathrm{Then}\:\mathrm{keep}\:\mathrm{in}\:\mathrm{mind}\:\mathrm{that}\:\mathrm{the}\:\mathrm{radius}\:\mathrm{and}\:\mathrm{the}\:\mathrm{tangent} \\ $$$$\mathrm{at}\:\mathrm{the}\:\mathrm{common}\:\mathrm{point}\:\mathrm{are}\:\bot\:\mathrm{to}\:\mathrm{each}\:\mathrm{other}. \\ $$$$\mathrm{So}\:\mathrm{find}\:\mathrm{the}\:\mathrm{slope}\:\mathrm{of}\:\mathrm{the}\:\mathrm{tangent}\:.\:\mathrm{And}\:\mathrm{then}….. \\ $$$$\mathrm{you}\:\mathrm{know}\:\mathrm{what}\:\mathrm{to}\:\mathrm{do}. \\ $$
Commented by Rio Michael last updated on 01/Jun/20
thank you sir.
$$\mathrm{thank}\:\mathrm{you}\:\mathrm{sir}. \\ $$

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