a straight line bdc touches the circle at d such that bd = 30 cm. dc = 7 cm. the other tangents bf and ce are drawn from band c to the circle and meet when produced at a making bac a right angle. calculate the length of af and radius of the circle.
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" Let .AF be x cm Here, BD = BE ... Tangents from the same external point are equal .*. BE = 30 cm Similarly, CD = CF CF = 7 cm CD = 7 cm and AE = AF = x cm In AABC, LA = 90° BC2= AB2+ AC2 ... Pythagoras theorem
(30 + 7)2= (30 + x)2+ (x + 7)2 ...1369 = 900 + 60x + /2+ /2+ 14x + 49 1369 = 2/2+ 74x + 949 2/2+ 74x + 949 — 1369 = 0 .'. 212+ 74x — 420 = 0 .'.x2+37x-210=0 .'. x2+ 42x — 5x — 210 = 0 x(x +42)— 5(x +42). 0 (x — 5) = 0 or (x + 42) = x = 5 or x = — 42 (Neglect) AF = 5 cm (2) In quadrilateral OEAF, LA = 90° LE = LF = 90° ... Radius at the point of contact is perpendicular to the tangent LEOF = 90° Also AE = AF OEAF is a square ...OE = AF = 5 cm "
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