Please help me solving this ques

Please help me solving this ques 16. MATHEMATICS four distinct circles with radii 3, r,. rm r, (r, , r2. r, > 3) touches all three fixed distinct lines in a plane, r,. are roots of equation x' —mx2 + 243x—n = O, m, n e R, then value of is:

Dear student,

The circle with radius 3 is the incircle of the triangle formed by the lines and r1,r2 and r3 represent the exradii.Now r1r2r3=Δ3s-as-bs-c=3Δ2==rs2We know r1=stanA2r1r2+r2r3+r1r3=s2tanA2tanB2+tanB2tanC2+tanC2tanA2=s2Also as  r1,r2 and r3  are the roots of the equation we getr1r2+r2r3+r1r3=243=s2s=243=93r=Δs=9Δ=273Area of triangle /Area of incircle=2739π=33πSo the triangle is equilateraln=3*243=729
Also x3-mx2+243x-729=0 has real roots 
3x2-2mx+243=0 has real  rootsi.e4m24×243×3m27If m=27 we getwe get r1=r2=r3=9So we getm+n79+m=27+729736=3Regards

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