128
Question
Let ABCD be a square of side of unit length. Let a circle C1 centered at A with unit radius is drawn. Another circle C2 which touches C1 and the lines AD and AB are tangent to it, is also drawn. Let a tangent line from the point C to the circle C2 meet the side AB at E. If the length of EB is α+√3β where α,β are integers, then α+β is equal to
Let ABCD be a square of side of unit length. Let a circle C1 centered at A with unit radius is drawn. Another circle C2 which touches C1 and the lines AD and AB are tangent to it, is also drawn. Let a tangent line from the point C to the circle C2 meet the side AB at E. If the length of EB is α+√3β where α,β are integers, then α+β is equal to
Open in App
Solution
(i) √2r+r=1
r=1√2+1r=√2−1(ii) CC2=2√2−2=2(√2−1)
From △CC2N=sinϕ=√2−12(√2−1)
⇒ϕ=30∘
(iii) In △ACE, from sine law,
AEsinϕ=ACsin105∘⇒AE=12×√2√3+1.2√2⇒AE=2√3+1=√3−1∴EB=1−(√3−1)=2−√3
Hence α=2,β=−1
⇒α+β=1
(i) √2r+r=1
r=1√2+1r=√2−1(ii) CC2=2√2−2=2(√2−1)
From △CC2N=sinϕ=√2−12(√2−1)
⇒ϕ=30∘
(iii) In △ACE, from sine law,
AEsinϕ=ACsin105∘⇒AE=12×√2√3+1.2√2⇒AE=2√3+1=√3−1∴EB=1−(√3−1)=2−√3
Hence α=2,β=−1
⇒α+β=1
Suggest Corrections
0
