InterviewSolution
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InterviewSolution
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A variable triangle `A B C`is circumscribed about a fixed circle of unit radius. Side `B C`always touches the circle at D and has fixed direction. If B and C vary insuch a way that (BD) (CD)=2, then locus of vertex Awill be a straight line.parallel to side BCperpendicular to side BCmaking an angle `(pi/6)`with BCmaking an angle `sin^(-1)(2/3)`with `B C`A. parallel to side BCB. perpendicular to side BCC. making an angle `(pi//6)` with BCD. making an angle `sin^(-1) (2//3)` with BC |
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Answer» Correct Answer - A `BD = (s -b), CD = (s-c)` `rArr (s-b) (s-c) =2` or `s(s-a) (s-b) (s-c) = 2 s(s -a)` or `Delta^(2) = 2s (s-a)` or `(Dleta^(2))/(s^(2)) = (2(s-a))/(s)` (using `Delta = rs`) or `r^(2) = (2(s-a))/(s)` or `(a)/(s)` = constant Now, `Delta = (1)/(2) aH_(a)`, where `H_(a)` is the distance of A from BC. Thus, `(Delta)/(2) = (1)/(2) (aH_(a))/(s) = 1 " or " H_(a) = (2s)/(a)` = constant Therefore, locus of A will be a straight line parallel to side BC |
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