Two circles touch externally. The sum of their areas is 41Ï€ square cm. If the distance between their centres is 9cm, then what is difference between their diameters
Let the centre of the two circles be c1 and c2, then c1c2=9cm (given)
If r1 and r2 denotes the radii. Then, ‌πr12+πr22=41π ⇒‌‌r12+r22=41 ‌ Also ‌‌‌r1+r2=9 ‌ Put ‌r2=9−r1‌ into Eq (i), we get ‌ ⇒‌‌‌ (ii) ‌ ⇒r12+(9−r1)2=41 ⇒r12+81+r12−18r1=41 ⇒‌‌2r12−18r1+40=0 ⇒‌‌r12−9r1+20=0 ⇒r12−4r1−5r1+20=0 ⇒r1(r1−4)−5(r1−4)=0 ⇒‌‌(r1−4)(r1−5)=0 ∴r1=4‌ or ‌5 ‌ When ‌r1=4 ⇒r2=9−r1=9−4=5cm ‌ When ‌r1=5 ⇒r2=9−r1=9−5=4cm ‌ For, ‌r1=4cm‌ and ‌r2=5cm ⇒d1=2×4=8cm ‌ and ‌d2=2×5=10cm ∴d2−d1=10−8=2cm ‌ Similarly, for ‌r1=5cm‌ and ‌r2=4cm‌, ‌ ‌ the difference ‌‌ between ‌‌ the ‌ ‌ diameters is ‌2cm‌. ‌