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Let-S-be-a-circle-with-centre-O-A-chord-AB-not-a-diameter-divides-S-into-two-regions-R-1-and-R-2-such-that-O-belongs-to-R-2-Let-S-1-be-a-circle-with-centre-in-R-1-touching-AB-at-X-and-S-inte




Question Number 19699 by Tinkutara last updated on 14/Aug/17
Let S be a circle with centre O. A chord  AB, not a diameter, divides S into two  regions R_1  and R_2  such that O belongs  to R_2 . Let S_1  be a circle with centre in  R_1 , touching AB at X and S internally.  Let S_2  be a circle with centre in R_2 ,  touching AB at Y, the circle S internally  and passing through the centre of S.  The point X lies on the diameter  passing through the centre of S_2  and  ∠YXO = 30°. If the radius of S_2  is 100  then what is the radius of S_1 ?
$$\mathrm{Let}\:{S}\:\mathrm{be}\:\mathrm{a}\:\mathrm{circle}\:\mathrm{with}\:\mathrm{centre}\:{O}.\:\mathrm{A}\:\mathrm{chord} \\ $$$${AB},\:\mathrm{not}\:\mathrm{a}\:\mathrm{diameter},\:\mathrm{divides}\:{S}\:\mathrm{into}\:\mathrm{two} \\ $$$$\mathrm{regions}\:{R}_{\mathrm{1}} \:\mathrm{and}\:{R}_{\mathrm{2}} \:\mathrm{such}\:\mathrm{that}\:{O}\:\mathrm{belongs} \\ $$$$\mathrm{to}\:{R}_{\mathrm{2}} .\:\mathrm{Let}\:{S}_{\mathrm{1}} \:\mathrm{be}\:\mathrm{a}\:\mathrm{circle}\:\mathrm{with}\:\mathrm{centre}\:\mathrm{in} \\ $$$${R}_{\mathrm{1}} ,\:\mathrm{touching}\:{AB}\:\mathrm{at}\:{X}\:\mathrm{and}\:{S}\:\mathrm{internally}. \\ $$$$\mathrm{Let}\:{S}_{\mathrm{2}} \:\mathrm{be}\:\mathrm{a}\:\mathrm{circle}\:\mathrm{with}\:\mathrm{centre}\:\mathrm{in}\:{R}_{\mathrm{2}} , \\ $$$$\mathrm{touching}\:{AB}\:\mathrm{at}\:{Y},\:\mathrm{the}\:\mathrm{circle}\:{S}\:\mathrm{internally} \\ $$$$\mathrm{and}\:\mathrm{passing}\:\mathrm{through}\:\mathrm{the}\:\mathrm{centre}\:\mathrm{of}\:{S}. \\ $$$$\mathrm{The}\:\mathrm{point}\:{X}\:\mathrm{lies}\:\mathrm{on}\:\mathrm{the}\:\mathrm{diameter} \\ $$$$\mathrm{passing}\:\mathrm{through}\:\mathrm{the}\:\mathrm{centre}\:\mathrm{of}\:{S}_{\mathrm{2}} \:\mathrm{and} \\ $$$$\angle{YXO}\:=\:\mathrm{30}°.\:\mathrm{If}\:\mathrm{the}\:\mathrm{radius}\:\mathrm{of}\:{S}_{\mathrm{2}} \:\mathrm{is}\:\mathrm{100} \\ $$$$\mathrm{then}\:\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{radius}\:\mathrm{of}\:{S}_{\mathrm{1}} ? \\ $$

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