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CBSE Class 9 Answered

1. what is the sum of all digits of all the integers from 1 to 1000000 ?
2. a triangle with sides 5 , 12 and 13 has both an inscribed and a circum-scribed circle . what is the distance beween the centers of those circle ?
3. let x and y be two digit integers such that y is obtained b reversing the digits of x . suppose x and y also satisfies xsq.-ysq. =msq.  for some positive integers m . what is the value of m+x+y ?
 
Asked by dkvyas2005 | 21 Jan, 2016, 11:16: PM
answered-by-expert Expert Answer
begin mathsize 11px style In space the space figure comma space ABC space is space the space right space triangle space with space sides space 5 comma space 12 space and space 13. straight I space is space the space incentre space and space straight O space is space the space circumcenter. Let space EB equals straight x comma space then space AE equals 5 minus straight x equals AF CF equals AC minus AF equals 13 minus open parentheses 5 minus straight x close parentheses equals 8 plus straight x CF equals 8 plus straight x equals CD BD equals BC minus CD equals 12 minus open parentheses 8 plus straight x close parentheses equals 4 minus straight x In space quadrilateral space EBDI comma space angle straight B equals angle straight E equals angle straight D equals 90 degree BE equals BD. space left parenthesis length space of space tangents space from space straight B space to space the space same space circle right parenthesis Hence comma space EBDI space is space straight a space square. space BE equals BD rightwards double arrow straight x equals 4 minus straight x rightwards double arrow 2 straight x equals 4 rightwards double arrow straight x equals 2 So comma space inradius equals EI equals ID equals IF equals 2..... left parenthesis 1 right parenthesis For space straight a space right space traingle comma space circumcenter space is space the space midpoint space of space the space hypotenuse. Hence comma space AO equals AC over 2 equals 13 over 2 equals 6.5 AF equals 5 minus straight x equals 5 minus 2 equals 3 Hence comma space FO equals AO minus AF equals 6.5 minus 3 equals 3.5..... left parenthesis 2 right parenthesis IFO space is space straight a space right space triangle space with space IF equals 2 space and space FO equals 3.5 space left parenthesis from space left parenthesis 1 right parenthesis space and space left parenthesis 2 right parenthesis right parenthesis Hence comma space the space distance space between space incenter space and space the space circumcenter equals IO equals square root of IF squared plus FO squared end root equals square root of 2 squared plus 3.5 squared end root equals square root of 16.25 end root almost equal to 4.03 end style
Answered by satyajit samal | 22 Jan, 2016, 01:15: PM

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