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If from a variable point P representingthe complex number z_(1) on the curve |z|=4, two tangentsare drawn to thecurve |z|=2, meeting it at points Q(z_(2)) and R(z_(3)), then which of the following statement(s) is(are) correct? |
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Answer» Triangle , PQR is isosceles.  `:.` From above figure , `cos ( anglePOR)=(OR)/(OP)(2)/(4)=(1)/(2)` `rArr angle POR=(pi)/(3)=anglePOQ rArr angle OPR= angle OPQ=30^(@)` `rArr angleQPR=60^(@) " ".....(1)` ALSO, in `DeltaPQR, PQ=PR "" .....(2)` `:.` From (1) and (2) , we get `DeltaPQR` is equilateral `rArr` (A) is INCORRECT. Also, PQOR are CONCYCLIC and `angleOQP and angleORP = 90^(@)`. So, circumcentre of `DeltaPQR` passes through O(0,0) and OP is diameter of it. So, circumcentre of `DeltaPQR` = mid POINT of OP `=((0+4 cos theta)/(2),(0+4 sin theta)/(2))=(2cos theta, 2 sin theta)` =centroid of `DeltaPQR` [As, `DeltaPQR` is equilateral.] `:.` The locus of centroid of `DeltaPQR` is |z|=2 `rArr` (B) isincorrect. Also, circumradius of `DeltaPQR=(OP)/(2)=(4)/(2)=2rArr` (C) is correct. As, `r=(R)/(2)=(2)/(2)=1` (As, `DeltaPQR` is equilateral.) `rArr` radius of circle inscribed in `DeltaPQR` is 1. `rArr` (D) is correct.] |
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